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Law of large numbers
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An illustration of the law of large numbers using a particular run of rolls of a single die. As the number of rolls in this run increases, the average of the values of all the results approaches 3.5. While different runs would show a different shape over a small number of throws (at the left), over a large number of rolls (to the right) they would be extremely similar.
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In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
The LLN is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the law only applies (as the name indicates) when a large number of observations is considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy).
Contents
1 Examples
2 History
3 Forms
3.1 Weak law
3.2 Strong law
3.3 Differences between the weak law and the strong law
3.4 Uniform law of large numbers
3.5 Borel's law of large numbers
4 Proof of the weak law
4.1 Proof using Chebyshev's inequality assuming finite variance
4.2 Proof using convergence of characteristic functions
5 See also
6 Notes
7 References
8 External links
Examples[edit]
For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability. Therefore, the expected value of a single die roll is
1 + 2 + 3 + 4 + 5 + 6 6 = 3.5 {\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5}
According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the precision increasing as more dice are rolled.
It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. For a Bernoulli random variable, the expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.)) is precisely the relative frequency.
For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1/2. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 1/2. In particular, the proportion of heads after n flips will almost surely converge to 1/2 as n approaches infinity.
Though the proportion of heads (and tails) approaches 1/2, almost surely the absolute difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number, approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows.
History[edit]
Diffusion is an example of the law of large numbers, applied to chemistry. Initially, there are solute molecules on the left side of a barrier (magenta line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container.
Top: With a single molecule, the motion appears to be quite random.
Middle: With more molecules, there is clearly a trend where the solute fills the container more and more uniformly, but there are also random fluctuations.
Bottom: With an enormous number of solute molecules (too many to see), the randomness is essentially gone: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. In realistic situations, chemists can describe diffusion as a deterministic macroscopic phenomenon (see Fick's laws), despite its underlying random nature.
The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials.[1] This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli.[2] It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713. He named this his "Golden Theorem" but it became generally known as "Bernoulli's Theorem". This should not be confused with Bernoulli's principle, named after Jacob Bernoulli's nephew Daniel Bernoulli. In 1837, S.D. Poisson further described it under the name "la loi des grands nombres" ("The law of large numbers").[3][4] Thereafter, it was known under both names, but the "Law of large numbers" is most frequently used.
After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev,[5] Markov, Borel, Cantelli and Kolmogorov and Khinchin. Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true.[6][7] These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law, in reference to two different modes of convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.[6]
Forms[edit]
Two different versions of the law of large numbers are described below; they are called the strong law of large numbers, and the weak law of large numbers. Stated for the case where X1, X2, ... is an infinite sequence of i.i.d. Lebesgue integrable random variables with expected value E(X1) = E(X2) = ...= µ, both versions of the law state that – with virtual certainty – the sample average
X ¯ n = 1 n ( X 1 + ⋯ + X n ) {\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})}
converges to the expected value
X ¯ n → μ for n → ∞ , {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\,\to \,\mu \qquad {\textrm {for}}\qquad n\to \infty ,\\{}\end{matrix}}}
(law. 1)
(Lebesgue integrability of Xj means that the expected value E(Xj) exists according to Lebesgue integration and is finite. It does not mean that the associated probability measure is absolutely continuous with respect to Lebesgue measure.)
An assumption of finite variance Var(X1) = Var(X2) = ... = σ2 < ∞ is not necessary. Large or infinite variance will make the convergence slower, but the LLN holds anyway. This assumption is often used because it makes the proofs easier and shorter.
Mutual independence of the random variables can be replaced by pairwise independence in both versions of the law.[8]
The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see Convergence of random variables.
Weak law[edit]
Simulation illustrating the law of large numbers. Each frame, a coin that is red on one side and blue on the other is flipped, and a dot is added in the corresponding column. A pie chart shows the proportion of red and blue so far. Notice that while the proportion varies significantly at first, it approaches 50% as the number of trials increases.
The weak law of large numbers (also called Khinchin's law) states that the sample average converges in probability towards the expected value[9]
X ¯ n → P μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 2)
That is, for any positive number ε,
lim n → ∞ Pr ( | X ¯ n − μ | > ε ) = 0. {\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |>\varepsilon \,\right)=0.}
Interpreting this result, the weak law states that for any nonzero margin specified, no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin.
As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in the series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by Chebyshev as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of the average of the first n values goes to zero as n goes to infinity.[7] As an example, assume that each random variable in the series follows a Gaussian distribution with mean zero, but with variance equal to 2 n / log  ( n + 1 ) . {\displaystyle 2n/\log(n+1).} At each stage, the average will be normally distributed (as the average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is asymptotic to n 2 / log  n {\displaystyle n^{2}/\log n} . The variance of the average is therefore asymptotic to 1 / log  n {\displaystyle 1/\log n} and goes to zero.
An example where the law of large numbers does not apply is the Cauchy distribution. Let the random numbers equal the tangent of an angle uniformly distributed between −90° and +90°. The median is zero, but the expected value does not exist, and indeed the average of n such variables has the same distribution as one such variable. It does not tend toward zero as n goes to infinity.
There are also examples of the weak law applying even though the expected value does not exist. See #Differences between the weak law and the strong law.
Strong law[edit]
The strong law of large numbers states that the sample average converges almost surely to the expected value[10]
X ¯ n → a.s. μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\bar {X}}_{n}\ {\xrightarrow {\text{a.s.}}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 3)
That is,
Pr ( lim n → ∞ X ¯ n = μ ) = 1. {\displaystyle \Pr \!\left(\lim _{n\to \infty }{\bar {X}}_{n}=\mu \right)=1.}
What this means is that as the number of trials n goes to infinity, the probability that the average of the observations is equal to the expected value will be equal to one.
The proof is more complex than that of the weak law.[11] This law justifies the intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average".
Almost sure convergence is also called strong convergence of random variables. This version is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). The strong law implies the weak law but not vice versa, when the strong law conditions hold the variable converges both strongly (almost surely) and weakly (in probability). However the weak law may hold in conditions where the strong law does not hold and then the convergence is only weak (in probability).[clarification needed]
To date it has not been possible to prove that the strong law conditions are the same as those of the weak law.[citation needed]
The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem.
The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for the average to converge almost surely on something (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that).[12]
If the summands are independent but not identically distributed, then
X ¯ n − E  [ X ¯ n ] → a.s. 0 , {\displaystyle {\bar {X}}_{n}-\operatorname {E} {\big [}{\bar {X}}_{n}{\big ]}\ {\xrightarrow {\text{a.s.}}}\ 0,}
provided that each Xk has a finite second moment and
∑ k = 1 ∞ 1 k 2 Var  [ X k ] < ∞ . {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} [X_{k}]<\infty .}
This statement is known as Kolmogorov's strong law, see e.g. Sen & Singer (1993, Theorem 2.3.10).
An example of a series where the weak law applies but not the strong law is when Xk is plus or minus k / log  log  log  k {\displaystyle {\sqrt {k/\log \log \log k}}} (starting at sufficiently large k so that the denominator is positive) with probability 1/2 for each.[12] The variance of Xk is then k / log  log  log  k . {\displaystyle k/\log \log \log k.} Kolmogorov's strong law does not apply because the partial sum in his criterion up to k=n is asymptotic to log  n / log  log  log  n {\displaystyle \log n/\log \log \log n} and this is unbounded.
If we replace the random variables with Gaussian variables having the same variances, namely k / log  log  log  k , {\displaystyle {\sqrt {k/\log \log \log k}},} then the average at any point will also be normally distributed. The width of the distribution of the average will tend toward zero (standard deviation asymptotic to 1 / 2 log  log  log  n {\displaystyle 1/{\sqrt {2\log \log \log n}}} ), but for a given ε, there is probability which does not go to zero with n that the average sometime after the nth trial will come back up to ε. Since this probability does not go to zero, it must have a positive lower bound p(ε), which means there is a probability of at least p(ε) that the average will attain ε after n trials. It will happen with probability p(ε)/2 before some m which depends on n. But even after m, there is still a probability of at least p(ε) that it will happen. (This seems to indicate that p(ε)=1 and the average will attain ε an infinite number of times.)
Differences between the weak law and the strong law[edit]
The weak law states that for a specified large n, the average X ¯ n {\displaystyle {\overline {X}}_{n}} is likely to be near μ. Thus, it leaves open the possibility that | X ¯ n − μ | > ε {\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon } happens an infinite number of times, although at infrequent intervals. (Not necessarily | X ¯ n − μ | ≠ 0 {\displaystyle |{\overline {X}}_{n}-\mu |\neq 0} for all n).
The strong law shows that this almost surely will not occur. In particular, it implies that with probability 1, we have that for any ε > 0 the inequality | X ¯ n − μ | < ε {\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon } holds for all large enough n.[13]
The strong law does not hold in the following cases, but the weak law does.[14][15][16]
1. Let X be an exponentially distributed random variable with parameter 1. The random variable sin  ( X ) e X X − 1 {\displaystyle \sin(X)e^{X}X^{-1}} has no expected value according to Lebesgue integration, but using conditional convergence and interpreting the integral as a Dirichlet integral, which is an improper Riemann integral, we can say:
E ( sin  ( X ) e X X ) = ∫ 0 ∞ sin  ( x ) e x x e − x d x = π 2 {\displaystyle E\left({\frac {\sin(X)e^{X}}{X}}\right)=\ \int _{0}^{\infty }{\frac {\sin(x)e^{x}}{x}}e^{-x}dx={\frac {\pi }{2}}}
2. Let x be geometric distribution with probability 0.5. The random variable 2 X ( − 1 ) X X − 1 {\displaystyle 2^{X}(-1)^{X}X^{-1}} does not have an expected value in the conventional sense because the infinite series is not absolutely convergent, but using conditional convergence, we can say:
E ( 2 X ( − 1 ) X X ) = ∑ 1 ∞ 2 x ( − 1 ) x x 2 − x = − ln  ( 2 ) {\displaystyle E\left({\frac {2^{X}(-1)^{X}}{X}}\right)=\ \sum _{1}^{\infty }{\frac {2^{x}(-1)^{x}}{x}}2^{-x}=-\ln(2)}
3. If the cumulative distribution function of a random variable is
1 − F ( x ) = e 2 x ln  ( x ) , x ≥ e {\displaystyle 1-F(x)={\frac {e}{2x\ln(x)}},x\geq e}
F ( x ) = e − 2 x ln  ( − x ) , x ≤ − e {\displaystyle F(x)={\frac {e}{-2x\ln(-x)}},x\leq -e}
then it has no expected value, but the weak law is true.[17][18]
Uniform law of large numbers[edit]
Suppose f(x,θ) is some function defined for θ ∈ Θ, and continuous in θ. Then for any fixed θ, the sequence {f(X1,θ), f(X2,θ), …} will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E[f(X,θ)]. This is the pointwise (in θ) convergence.
The uniform law of large numbers states the conditions under which the convergence happens uniformly in θ. If[19][20]
Θ is compact,
f(x,θ) is continuous at each θ ∈ Θ for almost all xs, and measurable function of x at each θ.
there exists a dominating function d(x) such that E[d(X)] < ∞, and
‖ f ( x , θ ) ‖ ≤ d ( x ) for all θ ∈ Θ . {\displaystyle \left\|f(x,\theta )\right\|\leq d(x)\quad {\text{for all}}\ \theta \in \Theta .}
Then E[f(X,θ)] is continuous in θ, and
sup θ ∈ Θ ‖ 1 n ∑ i = 1 n f ( X i , θ ) − E  [ f ( X , θ ) ] ‖ → a . s . 0. {\displaystyle \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta )-\operatorname {E} [f(X,\theta )]\right\|{\xrightarrow {\mathrm {a.s.} }}\ 0.}
This result is useful to derive consistency of a large class of estimators (see Extremum estimator).
Borel's law of large numbers[edit]
Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event occurs approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if E denotes the event in question, p its probability of occurrence, and Nn(E) the number of times E occurs in the first n trials, then with probability one,[21]
N n ( E ) n → p as n → ∞ . {\displaystyle {\frac {N_{n}(E)}{n}}\to p{\text{ as }}n\to \infty .}
This theorem makes rigorous the intuitive notion of probability as the long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory.
Chebyshev's inequality. Let X be a random variable with finite expected value μ and finite non-zero variance σ2. Then for any real number k > 0,
Pr ( | X − μ | ≥ k σ ) ≤ 1 k 2 . {\displaystyle \Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.}
Proof of the weak law[edit]
Given X1, X2, ... an infinite sequence of i.i.d. random variables with finite expected value E(X1) = E(X2) = ... = µ < ∞, we are interested in the convergence of the sample average
X ¯ n = 1 n ( X 1 + ⋯ + X n ) . {\displaystyle {\overline {X}}_{n}={\tfrac {1}{n}}(X_{1}+\cdots +X_{n}).}
The weak law of large numbers states:
Theorem: X ¯ n → P μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 2)
Proof using Chebyshev's inequality assuming finite variance[edit]
This proof uses the assumption of finite variance Var  ( X i ) = σ 2 {\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} (for all i {\displaystyle i} ). The independence of the random variables implies no correlation between them, and we have that
Var  ( X ¯ n ) = Var  ( 1 n ( X 1 + ⋯ + X n ) ) = 1 n 2 Var  ( X 1 + ⋯ + X n ) = n σ 2 n 2 = σ 2 n . {\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}
The common mean μ of the sequence is the mean of the sample average:
E ( X ¯ n ) = μ . {\displaystyle E({\overline {X}}_{n})=\mu .}
Using Chebyshev's inequality on X ¯ n {\displaystyle {\overline {X}}_{n}} results in
P  ( | X ¯ n − μ | ≥ ε ) ≤ σ 2 n ε 2 . {\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\leq {\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}
This may be used to obtain the following:
P  ( | X ¯ n − μ | < ε ) = 1 − P  ( | X ¯ n − μ | ≥ ε ) ≥ 1 − σ 2 n ε 2 . {\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|<\varepsilon )=1-\operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\geq 1-{\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}
As n approaches infinity, the expression approaches 1. And by definition of convergence in probability, we have obtained
X ¯ n → P μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 2)
Proof using convergence of characteristic functions[edit]
By Taylor's theorem for complex functions, the characteristic function of any random variable, X, with finite mean μ, can be written as
φ X ( t ) = 1 + i t μ + o ( t ) , t → 0. {\displaystyle \varphi _{X}(t)=1+it\mu +o(t),\quad t\rightarrow 0.}
All X1, X2, ... have the same characteristic function, so we will simply denote this φX.
Among the basic properties of characteristic functions there are
φ 1 n X ( t ) = φ X ( t n ) and φ X + Y ( t ) = φ X ( t ) φ Y ( t ) {\displaystyle \varphi _{{\frac {1}{n}}X}(t)=\varphi _{X}({\tfrac {t}{n}})\quad {\text{and}}\quad \varphi _{X+Y}(t)=\varphi _{X}(t)\varphi _{Y}(t)\quad } if X and Y are independent.
These rules can be used to calculate the characteristic function of X ¯ n {\displaystyle \scriptstyle {\overline {X}}_{n}} in terms of φX:
φ X ¯ n ( t ) = [ φ X ( t n ) ] n = [ 1 + i μ t n + o ( t n ) ] n → e i t μ , as n → ∞ . {\displaystyle \varphi _{{\overline {X}}_{n}}(t)=\left[\varphi _{X}\left({t \over n}\right)\right]^{n}=\left[1+i\mu {t \over n}+o\left({t \over n}\right)\right]^{n}\,\rightarrow \,e^{it\mu },\quad {\text{as}}\quad n\rightarrow \infty .}
The limit eitμ is the characteristic function of the constant random variable μ, and hence by the Lévy continuity theorem, X ¯ n {\displaystyle \scriptstyle {\overline {X}}_{n}} converges in distribution to μ:
X ¯ n → D μ for n → ∞ . {\displaystyle {\overline {X}}_{n}\,{\xrightarrow {\mathcal {D}}}\,\mu \qquad {\text{for}}\qquad n\to \infty .}
μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see Convergence of random variables.) Therefore,
X ¯ n → P μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 2)
This shows that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists.
See also[edit]
Asymptotic equipartition property
Central limit theorem
Infinite monkey theorem
Law of averages
Law of the iterated logarithm
Lindy effect
Regression toward the mean
Sortition
Notes[edit]
Jump up ^ Mlodinow, L. The Drunkard's Walk. New York: Random House, 2008. p. 50.
Jump up ^ Jakob Bernoulli, Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis, 1713, Chapter 4, (Translated into English by Oscar Sheynin)
Jump up ^ Poisson names the "law of large numbers" (la loi des grands nombres) in: S.D. Poisson, Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés (Paris, France: Bachelier, 1837), p. 7. He attempts a two-part proof of the law on pp. 139–143 and pp. 277 ff.
Jump up ^ Hacking, Ian. (1983) "19th-century Cracks in the Concept of Determinism", Journal of the History of Ideas, 44 (3), 455-475 JSTOR 2709176
Jump up ^ Tchebichef, P. (1846). "Démonstration élémentaire d'une proposition générale de la théorie des probabilités". Journal für die reine und angewandte Mathematik (Crelles Journal). 1846 (33): 259–267. doi:10.1515/crll.1846.33.259.
^ Jump up to: a b Seneta 2013.
^ Jump up to: a b Yuri Prohorov. "Law of large numbers". Encyclopedia of Mathematics.
Jump up ^ Etemadi, N.Z. (1981). "An elementary proof of the strong law of large numbers". Wahrscheinlichkeitstheorie verw Gebiete. 55 (1): 119–122. doi:10.1007/BF01013465.
Jump up ^ Loève 1977, Chapter 1.4, p. 14
Jump up ^ Loève 1977, Chapter 17.3, p. 251
Jump up ^ "The strong law of large numbers – What's new". Terrytao.wordpress.com. Retrieved 2012-06-09.
^ Jump up to: a b Yuri Prokhorov. "Strong law of large numbers". Encyclopedia of Mathematics.
Jump up ^ Ross (2009)
Jump up ^ Lehmann, Erich L; Romano, Joseph P (2006-03-30). Weak law converges to constant. ISBN 9780387276052.
Jump up ^ "A NOTE ON THE WEAK LAW OF LARGE NUMBERS FOR EXCHANGEABLE RANDOM VARIABLES" (PDF). Dguvl Hun Hong and Sung Ho Lee.
Jump up ^ "weak law of large numbers: proof using characteristic functions vs proof using truncation VARIABLES".
Jump up ^ Mukherjee, Sayan. "Law of large numbers" (PDF).
Jump up ^ J. Geyer, Charles. "Law of large numbers" (PDF).
Jump up ^ Newey & McFadden 1994, Lemma 2.4
Jump up ^ Jennrich, Robert I. (1969). "Asymptotic Properties of Non-Linear Least Squares Estimators". The Annals of Mathematical Statistics. 40 (2): 633–643. doi:10.1214/aoms/1177697731.
Jump up ^ An Analytic Technique to Prove Borel's Strong Law of Large Numbers Wen, L. Am Math Month 1991
References[edit]
Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes, 2nd Edition. Clarendon Press, Oxford. ISBN 0-19-853665-8.
Richard Durrett (1995). Probability: Theory and Examples, 2nd Edition. Duxbury Press.
Martin Jacobsen (1992). Videregående Sandsynlighedsregning (Advanced Probability Theory) 3rd Edition. HCØ-tryk, Copenhagen. ISBN 87-91180-71-6.
Loève, Michel (1977). Probability theory 1 (4th ed.). Springer Verlag.
Newey, Whitney K.; McFadden, Daniel (1994). Large sample estimation and hypothesis testing. Handbook of econometrics, vol. IV, Ch. 36. Elsevier Science. pp. 2111–2245.
Ross, Sheldon (2009). A first course in probability (8th ed.). Prentice Hall press. ISBN 978-0-13-603313-4.
Sen, P. K; Singer, J. M. (1993). Large sample methods in statistics. Chapman & Hall, Inc.
Seneta, Eugene (2013), "A Tricentenary history of the Law of Large Numbers", Bernoulli, 19 (4): 1088–1121, arXiv:1309.6488  , doi:10.3150/12-BEJSP12
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Law of large numbers", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Weak Law of Large Numbers". MathWorld.
Weisstein, Eric W. "Strong Law of Large Numbers". MathWorld.
Animations for the Law of Large Numbers by Yihui Xie using the R package animation
Apple CEO Tim Cook said something that would make statisticians cringe. "We don't believe in such laws as laws of large numbers. This is sort of, uh, old dogma, I think, that was cooked up by somebody [..]" said Tim Cook and while: "However, the law of large numbers has nothing to do with large companies, large revenues, or large growth rates. The law of large numbers is a fundamental concept in probability theory and statistics, tying together theoretical probabilities that we can calculate to the actual outcomes of experiments that we empirically perform. explained Business Insider
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Shrek The Musical
From Wikipedia, the free encyclopedia
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This article is about the stage musical. For the series, see Shrek (franchise).
Shrek
The Musical
Broadway Playbill cover
Music
Jeanine Tesori
Lyrics
David Lindsay-Abaire
Book
David Lindsay-Abaire
Basis
Shrek
by DreamWorks Animation
and Shrek!
by William Steig
Productions
2008 Seattle
2008 Broadway
2010 US Tour
2011 West End
2011 Madrid
2014 First UK Tour
2014 German Tour
2015 Buenos Aires
2016 Budapest
2016 Halifax, Nova Scotia
2018 Second UK Tour
Shrek The Musical is a musical with music by Jeanine Tesori and book and lyrics by David Lindsay-Abaire. It is based on the 2001 DreamWorks Animation's film Shrek and William Steig's 1990 book Shrek!. After a trial run in Seattle, the original Broadway production opened in December 2008 and closed after a run of over 12 months in January 2010. It was followed by a tour of the United States which opened in 2010, and a re-vamped West End production from June 2011 to February 2013. Since its debut, the musical's rights have been available for independent theaters overseas, who have chosen to stage their own versions of the show.
A high definition filming of the Broadway production was released on DVD, Blu-ray and digital download on October 15, 2013 in North America and December 2 in the United Kingdom.[1][2][3]
Contents
1 History
1.1 Development
1.2 Seattle premiere (2008)
1.3 Broadway production (2008–10)
1.4 US national tours
1.5 West End production (2011–13)
1.6 UK and Ireland tours
1.7 International productions
2 Synopsis of the Broadway production
2.1 Act I
2.2 Act II
3 Musical numbers
3.1 Broadway
3.2 US Tour
4 Casts
5 Instrumentation
6 Production references
7 Recordings
8 Reception
9 Awards and nominations
9.1 Original Broadway production
9.2 Original London production
10 Home media
11 References
12 External links
History[edit]
Development[edit]
Lindsay-Abaire and Jason Moore (director) began working on the show in 2002, with Tesori joining the team from 2004.[4] A reading took place on August 10, 2007, with Stephen Kramer Glickman in the role of Shrek, Celia Keenan-Bolger as Princess Fiona, Robert L. Daye, Jr. as Donkey and Christopher Sieber as Lord Farquaad.[5]
Seattle premiere (2008)[edit]
The musical premiered in an out-of-town tryout at the 5th Avenue Theatre in Seattle. Previews began August 14, 2008, with an opening night of September 10. The tryout ran through September 21, and played to generally favorable reviews, being cited as one of the few movie-to-stage adaptations "with heart".[6] The principal cast included Brian d'Arcy James as Shrek, Sutton Foster as Princess Fiona, Christopher Sieber as Lord Farquaad, Chester Gregory II as Donkey,[7] John Tartaglia as Pinocchio and Kecia Lewis-Evans as the Dragon.
During previews, "I Could Get Used to This" was replaced by "Don't Let Me Go," and "Let Her In" became "Make a Move". Also during previews, a brief reprise of "Who I'd Be" was sung after Shrek overhears Fiona's misleading comment about being with a hideous beast, which led into "Build a Wall". This was cut and "Build a Wall" was placed after "Morning Person (Reprise)". "Build a Wall" was later cut during previews, but re-instated towards the end of the run.
Broadway production (2008–10)[edit]
After extensive changes were made, the show began previews on Broadway at the Broadway Theatre on November 8, 2008, with the official opening on December 14. The cast included Brian d'Arcy James as Shrek, Sutton Foster as Fiona, Christopher Sieber as Farquaad and Tartaglia as Pinocchio. Daniel Breaker took over the role of Donkey, as the creative team thought Chester Gregory II did not fit the part.[8] The Dragon was voiced by company members Haven Burton, Aymee Garcia and Rachel Stern, instead of a soloist. Kecia Lewis-Evans, who played Dragon in Seattle, was offered a part in the show's ensemble but declined.[6] Ben Crawford was the standby for Shrek, until he replaced d'Arcy James for the final months of performances.
The song "I'm a Believer", which was originally played as the audience left the theatre, was added to the score on October 2, 2009, and sung by the entire company at the end of the performance.[9]
The Broadway production of the show received a total of twelve Drama Desk Award and eight Tony Award nominations, including Best Musical and acting awards for d'Arcy James, Foster and Sieber. At the Tony Awards, the entire cast performed a section of "Freak Flag" for the opening number medley; later on, d'Arcy James, Foster and Breaker introduced Sieber and company, who performed "What's Up Duloc?".
The Broadway production closed on January 3, 2010, after 441 performances and 37 previews. At the time, it was one of the most expensive musicals to open on Broadway, at an estimated $25 million, and despite generally good reviews, it failed to recoup its initial investment.[10] The show was then extremely modified for the national tour.
US national tours[edit]
A national tour of North America began previews at the Cadillac Palace Theatre in Chicago, on July 13, 2010, with opening night on July 25.[11][12] Rob Ashford is the co-director, as the Broadway creative team revised changes. The production marked the debut of an all-new Dragon, voiced off-stage by a single vocalist, with four puppeteers controlling the movements of the new 25-foot puppet.. On the subject, set designer Tim Hatley stated "The biggest change [will be] the dragon. It will be a different creature from the puppet/soul trio on Broadway [but] I think we've finally gotten it right".[13] The tour also features a new opening, new songs and improved illusions, from those on Broadway.[14]
Many changes made for the tour include a new song sung by the dragon entitled "Forever", replacing "Donkey Pot Pie".
The original touring cast featured Eric Petersen as Shrek, Haven Burton as Princess Fiona, Alan Mingo, Jr. as Donkey, and David F.M. Vaughn as Lord Farquaad. Carrie Compere played the Dragon, with Blakely Slaybaugh as Pinocchio. Todd Buonopane was originally cast in the role of Lord Farquaad, but was replaced by Vaughn before opening. [15] The tour played its final performance at the Pantages Theatre in Los Angeles on July 31, 2011, ahead of a non-equity tour in September.[16]
A second tour of North America, featuring a Non-Equity cast, launched September 9, 2011, at the Capitol Theatre in Yakima, Washington.[17] Merritt David Janes appeared as Lord Farquaad.[18] The tour officially opened in Portland, Oregon on September 13, 2011. The tour ran in the U.S. through April 29, 2012, with the final show in Springfield, Missouri,[17] before playing Asia.
The second non-equity tour began October 5, 2012, in Anchorage, Alaska, ending on April 7, 2013, in Reno, Nevada.[19]
West End production (2011–13)[edit]
A newly revised scaled down version, began performances in the West End at the Theatre Royal Drury Lane, on May 6, 2011.[20] Nigel Lindsay headlined as Shrek, Richard Blackwood as Donkey, Nigel Harman as Lord Farquaad and Amanda Holden as Princess Fiona. Landi Oshinowo plays the Dragon, with Jonathan Stewart as Pinocchio.[21][22][23][24] The official opening night took place on June 14, 2011. Most critics were positive about the production, and in particular praised Harman's performance, branding him "hysterically funny".[25]
The show was nominated for a total of four awards at the 2012 Laurence Olivier Awards, including Best New Musical, Best Actor for Lindsay and Supporting Actor for Harman, as well as Best Costume Design for Tim Hatley. Harman won the award for Best Performance in a Supporting Role in a Musical for his performance as Lord Farquaad. The ensemble cast performed "Freak Flag" at the awards.
Kimberley Walsh, of UK pop group Girls Aloud, took over the role of Princess Fiona from October 5, 2011, after Holden announced her pregnancy. Dean Chisnall and Neil McDermott took over from Lindsay and Harman as Shrek and Lord Farquaad respectively on February 29, 2012.[26] Carley Stenson later took over as Princess Fiona from May 23, 2012.[27]
The London production of the show came to an end after 715 performances, on February 24, 2013. Producers announced their plans to tour Shrek across the UK in 2014.[28]
UK and Ireland tours[edit]
The first UK and Ireland tour began at the Grand Theatre in Leeds, England on July 23, 2014, before touring across the UK and Ireland. Dean Chisnall repeats his West End performance as Shrek, under the direction of Nigel Harman, who originated the role of Lord Farquaad in the West End.[29] A full company announcement was made in February 2014, with Chisnall to be joined by Legally Blonde star Faye Brookes as Princess Fiona, Gerard Carey as Lord Farquaad, Idriss Kargbo as Donkey, Candace Furbert as Dragon and Will Haswell as Pinocchio.[30] A cast change for the tour took place July 8, 2015, with ensemble member Bronté Barbé taking over the role of Princess Fiona from Brookes. The tour concluded at The Lowry, Salford on February 20, 2016.
A second UK and Ireland tour commenced at the Edinburgh Playhouse from 12 December 2017. Nigel Harman once again directs the tour. The tour is currently scheduled to run until at least January 2019. The full cast was announced in November 2017 [31] The X Factor star Amelia Lily and Call the Midwife actress Laura Main will share the role of Princess Fiona, alongside Samuel Holmes as Lord Farquaad, Stefan Harri as Shrek and Marcus Ayton as Donkey
International productions[edit]
Shrek was made available for independent US and overseas theatres. They have chosen to stage their own versions of the show with the same music, book and lyrics intact and their own designs for sets, costuming and other creative elements. Productions have been staged in Asia, Poland, Spain, France, Italy, Germany, The Netherlands, Belgium, Mexico, Brazil, the Philippines, Estonia, Finland, Israel, Sweden, Panama, Denmark, Canada, Puerto Rico, Argentina and Norway.
Synopsis of the Broadway production[edit]
Act I[edit]
On his seventh birthday, two ogre parents send their son Shrek out of their house and into the world to make his living. They warn him that because of his looks, humans will hate him, and the last thing he will see is an angry mob before he dies. ("Big Bright Beautiful World"). Some years later, an embittered, grown up Shrek is living contentedly alone in a swamp. However, his solitude is disrupted when a band of fairytale creatures show up on his property, including:
Pinocchio;
The Three Bears;
The Three Little Pigs;
a Wicked Witch;
The Big Bad Wolf;
Peter Pan;
Ugly Duckling;
Fairy Godmother;
The White Rabbit;
Mad Hatter;
Sugar Plum Fairy;
Humpty Dumpty;
an Elf.
They explain of their banishment from the Kingdom of Duloc, by order of the evil Lord Farquaad, who exiled them for being freaks, under penalty of death if they ever return ("Story of My Life"). Even so, Shrek decides to travel to see Farquaad and try to regain his swamp, along with getting the Fairytale Creatures their homes back (but mostly to get his swamp back), with much encouragement from Pinocchio and the gang ("The Goodbye Song").
Along the way, Shrek rescues a talkative donkey from some of Farquaad's guards. In return for rescuing him, and offering his friendship, Donkey insists on tagging along to show him the way to Duloc ("Don't Let Me Go"), which Shrek reluctantly agrees to, due to him being lost.
Meanwhile, in the Kingdom of Duloc, Farquaad is torturing Gingy the Gingerbread Man into revealing the whereabouts of other Fairytale Creatures that are still hiding in the Kingdom so he can have them arrested as well. They are interrupted by one of his henchmen, Thelonious, who reveals that Farquaad's guards have acquired the Magic Mirror. Farquaad then sends Gingy to the swamp with all the other Fairytale Creatures. The Mirror reveals that Princess Fiona is currently trapped in a castle surrounded by hot boiling lava and guarded by a terrible fire-breathing dragon. Accepting this task, Farquaad decides to marry her to become king, and rushes out to prepare for the wedding before the Mirror can tell him what happens to Fiona at night. The Mirror then shows the audience the story of Fiona's childhood.
A seven-year-old Fiona dreams of the brave knight who, as her storybooks tell her, will one day rescue her from her tower and end her mysterious curse with "True Love's First Kiss". As she grows into a teenager, and then a headstrong woman, she becomes a little bit stir-crazy, but she never loses her faith in her fairytales ("I Know It's Today").
Shrek and Donkey arrive in Duloc where Farquaad expresses his love for his kingdom, accompanied by his cheerful cookie-cut army of Duloc Dancers ("Welcome to Duloc" / "What's Up, Duloc?"). They approach Farquaad, with him being impressed by Shrek's size and appearance. Farquaad demands that Shrek must rescue Fiona, and in return, he will give Shrek the deed to his swamp ("What's Up, Duloc? (Reprise)").
The two unlikely friends set off to find Fiona, with Shrek becoming increasingly annoyed with Donkey as time progresses ("Travel Song"). After crossing the rickety old bridge and arriving at the castle, Shrek sets off alone to rescue Fiona while Donkey encounters a ferocious female dragon who initially wants to eat him, but then wants to keep him for her own after Donkey manages to charm her ("Donkey Pot Pie")/("Forever"). When Shrek finds Fiona, his lack of interest in playing out her desired, romantic rescue scene annoys her, and he drags her off by force. The two of them reunite with Donkey, and all three attempt to escape while being chased by the angry Dragon. Shrek traps Dragon and they get to a safe point ("This Is How A Dream Comes True").
Fiona then insists that Shrek reveal his identity and is shocked that her rescuer is an ogre and not the Prince Charming her stories indicated. Shrek explains that he is merely her champion; instead, she is to marry Farquaad. The trio begin their journey back to Duloc, but Fiona becomes apprehensive as the sun begins to set. She insists that they rest for the night and that she spend the night alone in a nearby cave. Donkey and Shrek remain awake, with Donkey asking Shrek who he would be, if he did not have to be an ogre anymore. As Shrek opens up to Donkey on who he would wish to be, Fiona transforms into an ogress as part of her curse that happens during sunset, stands apart, alone, and listens ("Who I'd Be").
Act II[edit]
The next day, Princess Fiona rises early and sings with a bluebird and dances with a deer (before making the bird explode and throwing the deer off a cliff). She assists the Pied Piper in his rat-charming duties ("Morning Person"). Shrek brings down her mood by attempting to give subtle hints about her groom-to-be ("Men of Farquaad's stature are in short supply", "He's very good at small talk", etc.) and mocking her tragic childhood circumstances. The two begin a contest of trying to one-up each other to outdo the others' backstory, but end up revealing their respective pasts ("I Think I Got You Beat"). Both admit to being thrown out by their parents; this connection, as well as bonding over a love of disgusting bodily noises, kindles friendship.
Back in Duloc, Lord Farquaad plans his wedding, and he reveals his own sordid heritage after The Magic Mirror insists that Farquaad should invite his father, but Farquaad refuses, explaining how he abandoned him in the woods as a child ("The Ballad of Farquaad").
As Shrek and Fiona's newfound camaraderie grows into love, Donkey insists, with the help of the Three Blind Mice from his imagination, that Shrek should gather his courage and romantically engage Fiona ("Make a Move"). Shrek, finally beginning to come out of his caustic, protective shell, tries to find the words to explain his feelings to Fiona ("When Words Fail").
While Shrek is out finding a flower for Fiona, Donkey discovers that Fiona turns into an ogress at night, and she confesses that she was cursed as a child, which is why she was locked away in the tower. Only a kiss from her true love will return her to her proper form, and she asks Donkey to promise never to tell. Shrek arrives near the end of the conversation and misunderstands Fiona's description of herself as an ugly beast, and thinks she is talking about him. Hurt by her presumed opinion, Shrek storms off.
The next day, transformed back to her human form, Fiona decides to tell Shrek about her curse ("Morning Person (Reprise)"). When she tries to explain, Shrek rebuffs her with his "ugly beast" overhearing, causing Fiona in turn to misunderstand. Then Farquaad arrives to claim Fiona and tells Shrek he has cleared the swamp of the Fairytale Creatures, and now belongs to Shrek again. While not very impressed with Farquaad, Fiona agrees to marry him and insists that they have the wedding before sunset. As Farquaad and Fiona ride back to Duloc, Donkey tries to explain the misunderstanding to Shrek (who is too angry and upset to listen), and Shrek rejects him as well, declaring that he will return to his swamp alone and build a wall to shield himself from the world ("Build a Wall").
Meanwhile, the Fairytale Creatures are on their way to a landfill which is to be their new home, since they have been evicted from the swamp. After dealing with the fact that Shrek broke his promise to them, they start agreeing that Farquaad's treatment of them is intolerable; just because they are freaks does not mean they deserve to be hated, and start deciding on doing something about it. But a bitter Pinocchio (remembering they are not allowed back in the Kingdom, or they will be executed), not wanting his friends to get hurt, suggests they should just keep going, and wait until everything gets better, all the while, wishing to be a real boy. The gang convince Pinocchio that they need to stand up to Farquaad, while also inspiring him to accept who he is, as all of them have accepted who they are. They gather new confidence and strength in themselves, as they declare that they'll raise their "Freak Flag" high against their tormentors ("Freak Flag"). Now realizing that they have become something more than friends, and have become a family, Pinocchio and the gang now set off to Duloc to stand up to Farquaad once and for all.
Shrek has returned to his once again private swamp, but he misses Fiona. Donkey shows up attempting to seal off his half of the swamp with stone boulders, which Shrek rebuffs. In turn, Donkey angrily berates Shrek for his reclusive and stubborn habits, even to the point of driving off Fiona. An angered Shrek reveals he heard her talking about a hideous creature the night before, and Donkey retorts that they were not talking about him, but of "someone else". When a confused Shrek inquires who it was, Donkey, wanting to keep his promise, and still cross with Shrek, refuses to talk. When Shrek apologizes and extends his friendship, Donkey forgives him.
The two then go back to Duloc, where Shrek interrupts the wedding before Farquaad can kiss Fiona, and Fiona convinces him to let Shrek speak with her. Shrek finally finds the words to express his feelings for Fiona, and he declares his love for her ("Big Bright Beautiful World (Reprise)"). However, his declaration of love is mocked by Farquaad. Caught between love and her desire to break the curse, Fiona tries to escape the event. Just then, the Fairytale Creatures storm into the wedding and protest their banishment. They are accompanied by Grumpy, one of the Seven Dwarfs, who reveals that he is Farquaad's father and he kicked Farquaad out at the age of 28 when he wouldn't move out of the basement, revealing Farquaad is a freak as well. During the argument, the sun sets, causing Fiona to turn into an ogress in front of everyone. Farquaad, furious and disgusted over the change, orders for Shrek to be drawn and quartered (along with the Fairytale Creatures) and Fiona banished back to her tower. As Farquaad proclaims himself the new King, Shrek whistles for the Dragon, who has now escaped the castle (and is the reason Shrek and Donkey got to the wedding just in time). Dragon then crashes through the window with Donkey and incinerates Farquaad with her fiery breath.
With Farquaad dead, Shrek and Fiona admit their love for each other and share true love's first kiss. Fiona's curse is broken, and she takes her true form: an ogress. At first, she is ashamed of her looks, but Shrek declares that she is still beautiful. The two ogres begin a new life together (along with Donkey and the Fairytale Creatures), as everyone celebrates what makes them special ("This Is Our Story") and they all live happily ever after ("I'm A Believer").
Musical numbers[edit]
Broadway[edit]
Act I
"Overture" – Orchestra
"Big Bright Beautiful World" – Mama Ogre, Papa Ogre, Shrek
"Story Of My Life" – Pinocchio, Elf, Peter Pan, Ugly Duckling, Fairy Godmother, Three Pigs, Wicked Witch, Mama Bear, Mad Hatter, Big Bad Wolf, Humpty Dumpty, White Rabbit, Three Bears, Fairytale Creatures
"Story Of My Life (Tag)" – Elf, Fairy Godmother, Pinocchio, Big Bad Wolf, Fairytale Creatures≠
"The Goodbye Song" – Elf, Fairytale Creatures≠
"Don't Let Me Go" – Donkey*
"Regiment" – Guards≠
"I Know It's Today" – Young Fiona, Teen Fiona, Fiona
"What's Up, Duloc?" – Lord Farquaad, Duloc Performers
"What’s Up, Duloc? (Reprise)" – Lord Farquaad, Duloc Performers≠
"Travel Song" – Donkey, Shrek
"Donkey Pot Pie" – Dragon, Donkey, Knights
"This Is How A Dream Comes True" – Fiona, Shrek, Donkey, Dragon
"Who I'd Be" – Shrek, Fiona, Donkey
Act II
"Entr’acte" – Orchestra≠
"Morning Person" – Fiona, Bluebird
"I Think I Got You Beat" – Fiona, Shrek
"The Ballad Of Farquaad" – Lord Farquaad, Guards
"Air Guitar Crossover" – Fiona, Shrek≠
"Make A Move" – Donkey, Three Blind Mice
"When Words Fail" – Shrek
"Morning Person (Reprise)" – Fiona
"The Arrival Of Farquaad" – Lord Farquaad
"Build A Wall" – Shrek
"Freak Flag" – Gingy, Mama Bear, Pinocchio, Three Pigs, Humpty Dumpty, Papa Bear, Elf, Wicked Witch, Mad Hatter, Big Bad Wolf, Fairytale Creatures
"Wedding Procession" – Choir≠
"Big Bright Beautiful World (Reprise)" – Shrek
"Cathedral Sunset/Transformation" – Crowd≠
"Beautiful Ain’t Always Pretty" – Shrek≠
"Finale" – Fiona, Shrek, Donkey, Mama Bear, Pinocchio, Gingy, Fairytale Creatures
"I'm a Believer" – Shrek, Three Pigs, Fiona, Young Fiona, Donkey, Full Company≠ (as of October 2, 2009)
≠ Not included on the original Broadway cast recording. "I'm a Believer" and "Forever", however, was recorded later and released as a single as it was not in the show when the cast recording was made.
Changes
An Overture and Entr'acte were added following the Seattle run, along with the song "Build a Wall" which was originally "More To The Story." This song was cut prior to the West End production opening.
"This Is Our Story" or "Finale" replaced "I Smell a Happy Ending" following the Seattle run, whilst "What Happens to Love?" and "More To The Story" were cut.
Beginning with the first North American tour, a new song was written for Dragon, "Forever." This was then performed in all subsequent productions, replacing "Donkey Pot Pie."
"Don't Let Me Go" was cut from the West End production, although an alternative version reinstated for the UK Tour.
US Tour[edit]
Act I
"Overture" – Orchestra
"Big Bright Beautiful World" – Mama Ogre, Papa Ogre, Queen Lillian, King Harold, Shrek, Happy People/Angry Mob
"Story Of My Life" – Pinocchio, Elf, Peter Pan, Ugly Duckling, Sugar Plum Fairy, Three Pigs, Wicked Witch, Mama Bear, Elf, Big Bad Wolf, Humpty Dumpty, Fairytale Creatures
"Story Of My Life (Tag)" – Elf, Ugly Duckling, Pinocchio, Big Bad Wolf, Fairytale Creatures≠
"The Goodbye Song" – Baby Bear, Fairytale Creatures≠
"Don't Let Me Go" – Donkey
"Regiment #1" – Guards≠
"Regiment (Reprise)" – Guards≠
"What's Up, Duloc?" – Lord Farquaad, Duloc Performers
"What’s Up, Duloc? (Reprise)" – Lord Farquaad, Duloc Performers≠
"I Know It's Today" – Young Fiona, Teen Fiona, Fiona
"Travel Song" – Donkey, Shrek
"Forever" – Dragon, Donkey, Knights
"This Is How A Dream Comes True" – Fiona, Shrek, Donkey, Dragon
"Who I'd Be" – Shrek, Fiona, Donkey
Act II
"Entr'acte" – Orchestra≠
"Morning Person" – Fiona, Bluebird
"I Think I Got You Beat" – Fiona, Shrek
"The Ballad Of Farquaad" – Lord Farquaad, Guards
"Air Guitar Crossover" – Fiona, Shrek≠
"Make A Move" – Donkey, Three Blind Mice
"Make A Move (Tag)" – Donkey, Three Blind Mice≠
"When Words Fail" – Shrek
"Morning Person (Reprise)" – Fiona
"The Arrival Of Farquaad" - Lord Farquaad
"Build A Wall" – Shrek
"Freak Flag" – Gingy, Sugar Plum Fairy, Pinocchio, Three Pigs, Ugly Duckling, Papa Bear, Fairy Godmother, Wicked Witch, Big Bad Wolf, Humpty Dumpty, Mama Bear, Fairytale Creatures
"Wedding Procession" – Choir≠
"Big Bright Beautiful World (Reprise)" – Shrek
"Cathedral Sunset/Transformation" – Crowd≠
"Beautiful Ain’t Always Pretty" – Shrek≠
"Finale" – Fiona, Shrek, Donkey, Mama Bear, Pinocchio, Gingy, Fairytale Creatures
"I'm a Believer" – Shrek, Three Pigs, Fiona, Young Fiona, Donkey, Dragon, Full Company≠ (as of October 2, 2009)
≠ Not included on the original Broadway cast recording. "I'm a Believer" and "Forever", however, was recorded later and released as a single as it was not in the show when the cast recording was made.
Casts[edit]
The original principal casts of the English-speaking productions.
Characters
Seattle
Broadway
U.S. Tour
West End
U.K. Tour
Budapest
Denmark
Shrek
Brian d'Arcy James Eric Petersen Nigel Lindsay Dean Chisnall Pál Feke Niels Olsen
Princess Fiona
Sutton Foster Haven Burton Amanda Holden Faye Brookes Dóra Szinetár Maria Lucia Heiberg Rosenberg
Donkey
Chester Gregory Daniel Breaker Alan Mingo, Jr. Richard Blackwood Idriss Kargbo András Csonka Alexandre Willaume
Lord Farquaad
Christopher Sieber David F.M. Vaughn Nigel Harman Gerard Carey Attila Serbán Martin Brygmann
Pinocchio
John Tartaglia Blakely Slaybaugh Jonathan Stewart Will Haswell Levente Juhász Silas Holst
Dragon
Kecia Lewis-Evans Haven Burton, Aymee Garcia, Rachel Stern Carrie Compere Landi Oshinowo Candace Furbert Gabi Tóth Szhirley
Gingy
Haven Burton Aymee Garcia Alice Fearn Nikki Bentley Kristine Yde Eriksen
Instrumentation[edit]
The Orchestra includes one bass guitar player, one trumpeter, one trombonist, two guitar players, one drummer, two violinists, two reed players, one horn player, two keyboard players, a cello player, and a percussion player. The guitar players double on ukulele, mandolins, electric guitars, and acoustic guitars. The trumpeter doubles on a flugelhorn and a piccolo trumpet. The trombonist doubles on tenor and bass trombones. The bass player doubles on the upright bass, the electric bass, and the 5-string bass guitar. The first reed doubles on alto sax, clarinet, flute, and piccolo. The second reed doubles on soprano sax, baritone sax, tenor sax, flute, bass clarinet, and clarinet.
The original Broadway orchestration included an additional trumpet, an additional trombone, two more violinists, one more cellist, and two more reed players. In this orchestration, the first reed doubles on piccolo, flute, and recorder. The second reed doubles on oboe, English horn, clarinet, and alto sax. The third reed doubles on flute, clarinet, bass clarinet, soprano sax, and tenor sax. The fourth reed doubles on clarinet, bassoon, and baritone sax.
Production references[edit]
This article contains a list of miscellaneous information. Please relocate any relevant information into other sections or articles. (April 2017)
Puss in Boots makes a silent cameo appearance during the "Travel Song". He tips his hat to Shrek and Donkey and walks away. Later, a bunny screams after Shrek calls him delicious. A cow jumps over the moon, and a dish runs away with a spoon, while being pursued by the police. Shrek and Donkey later see a giraffe and gazelles which look like the gazelle wheel in The Lion King (an excerpt from "Circle of Life" plays during this part of the song).
As with the 2001 film, Shrek ends the song "Travel Song" with "That'll do, Donkey, that'll do"; this is a reference to the similar phrase in the film Babe, in which Hoggett reassures "That'll do, Pig. That'll do."
King Harold and Queen Lillian originally sang a part during "Big Bright Beautiful World," explaining to Young Fiona that all ogres are monsters unloved by everyone. Now they walk past Young Shrek, taking Young Fiona with them. She waves at Shrek, but they lead her away (the guards preceding them give the impression that they are taking Fiona to her tower). This was later cut out of the Broadway production. A similar part was re-introduced for the national tour and London productions.
In "I Know It's Today" and "This is How A Dream Comes True," Fiona sings "Are you there, God, it's me, Fiona?" which is similar to the book entitled "Are You There, God? It's Me, Margaret" by Judy Blume.
In "Morning Person," much of the part where Fiona dances with the mice is reminiscent of the nightclub scene in Bob Fosse's "Sweet Charity."
Most of the spoken lines are taken from the first film, including the Muffin Man scene between Farquaad and Gingy, and Fiona explaining the curse to Donkey.
"Welcome to Duloc" is the only original song from the film sung in the musical. It is now "What’s Up, Duloc" and is sung by the Duloc Dancers, who resemble the puppets that sing it in the film.
Originally the orchestra played "I'm a Believer" after the curtain call, but as of October 2, 2009, it is sung by the entire company at the end of the performance.
At the end of "What's up, Duloc?" (Reprise), Lord Farquaad references Defying Gravity from the Broadway musical Wicked, when he sings "...and no ones gonna bring me down!" followed by the famous ending note. Also, the oft-repeated line 'Hey-nonny-nonny-nonny-no' is taken from the song "Shy" from Once Upon a Mattress.
At the end of "Forever", the Dragon's final line is a reference to the Dreamgirls song "And I Am Telling You I'm Not Going". In some adaptations of this song in later productions, the Dragon performer appears alongside the puppet (in previous performances, the performer sang off-stage.)
Julie Andrews, who played Queen Lillian in Shrek 2, Shrek the Third, and Shrek Forever After, provides the voice for the audio instructions before each performance that reminds the audience to turn off their cell phones, the use of recording devices is forbidden, etc. and "if you refuse, a terrifying ogre will leap from the stage, lift you from your seat, and drag you far, far away."
During "Story of My Life", one of the show's musical numbers, Mama Bear sings "Mama's in the mud, Mama's in distress," which is a nod to the song "Rose's Turn" from the musical Gypsy.
During "Freak Flag" a flag is waved around with a face on it, which resembles the famous logo of young Cosette from Les Misérables.
During a part of "Don't Let Me Go" on the national tour, Donkey sings to the music of "Take Me or Leave Me" and "I'll Cover You" from Rent. He also makes a reference to "La Vie Boheme" from the same musical while chanting things and marching across the stage in the bridge of the song.
During "What's Up, Duloc?," the ensemble reference the musical Chicago.
During a part of "Don't Let Me Go" original Broadway recording, Donkey makes reference to "Part Of Your World" from Little Mermaid.
Donkey sings to Shrek "Don't speak, don't speak, don't speak," referencing Dianne Wiest and John Cusack in "Bullets Over Broadway."
The opening of "Who I'd Be" is reminiscent of "Bring Him Home" from 'Les Misérables.
Recordings[edit]
The original Broadway cast recording was recorded on January 12, 2009, and was released on March 24, 2009, by Decca Broadway Records.[32] The album debuted at #1 on Billboard's Top Cast Albums chart and #88 on the Billboard 200.[33][34] "I'm a Believer" was not featured on the initial recording as it was only added to the show on October 2, 2009. It was later included as part of a Highlighted Cast Recording, released on November 17, 2009. On December 4, 2009, when the Grammy Award nominees were announced, the cast recording was nominated for Best Musical Show Album.[35]
"Donkey Pot Pie" (which is included on the original Broadway cast recording) was cut from future productions, replaced by "Forever." The song became available on iTunes in 2011. It was recorded during a live performance of the national tour in Chicago, and features Carrie Compere (Dragon) and Alan Mingo, Jr. (Donkey).
The original London cast recorded a single of "I'm a Believer" for promotional purposes.
An original Spanish-language cast recording featuring the Madrid cast was recorded between August and September 2011, and released in September. The Spanish album includes later added songs "Forever" and "I'm a Believer", as well as different orchestrations to the Broadway recording and the arrangements made for the national tour.
Reception[edit]
The musical has received mixed to positive reviews from critics. Ben Brantley wrote in The New York Times: "'Shrek,' for the record, is not bad.... As the title character, a misanthropic green ogre who learns to love, the talented Mr. James is... encumbered with padding and prosthetics.... As the evil, psychologically maimed Lord Farquaad, the very droll Christopher Sieber is required to walk on his knees, with tiny fake legs dangling before him — an initially funny sight gag that soon drags". He praises Sutton Foster as "an inspired, take-charge musical comedian.... Ms. Foster manages both to make fun of and exult in classical musical-comedy moves while creating a real, full character at the same time."[36]
Variety noted that the production had a reported budget of $24 million. Any "theme-park cutesiness is offset by the mischievous humor in David Lindsay-Abaire's book and lyrics. The production's real achievement, however, is that the busy visuals and gargantuan set-pieces never overwhelm the personalities of the actors or their characters. The ensemble is talented and the four leads, in particular, couldn't be better."[37]
The Associated Press said that "the folks at DreamWorks have done their darndest to make sure we are entertained at Shrek the Musical, the company's lavish stage adaptation of its hit animated movie. For much of the time, they succeed, thanks to the talent and ingratiating appeal of the show's four principal performers. The show's massive sets and colorful costumes (both courtesy of Tim Hatley) are so visually eye-catching that they often distract from what's going on with the story and score. Composer Jeanine Tesori has written attractive, eclectic, pop-flavored melodies that range from a jaunty 'Travel Song' to a gutsy duet called 'I Got You Beat' for Shrek and Fiona that revels in rude noises." The review also noted that Lindsay-Abaire's lyrics are often fun and quite witty.[38]
USA Today gave the show three and ½ out of four stars, writing: "Shrek, which draws from William Steig's book about a lovable ogre and the DreamWorks animated movie that it inspired, is nonetheless a triumph of comic imagination with a heart as big and warm as Santa's. It is the most ingeniously wacky, transcendently tasteless Broadway musical since The Producers, and more family-friendly than that gag-fest." The review also noted, however, that "Like other musical adaptations of hit films, Shrek... leans heavily on winking satire. There are the usual nods to more fully realized shows, from Gypsy to A Chorus Line, and Jeanine Tesori's blandly ingratiating score doesn't feature any songs you're likely to be humming 20 years from now."[39]
Awards and nominations[edit]
Original Broadway production[edit]
Year
Award Ceremony
Category
Nominee
Result
Ref
2009 Tony Award Best Musical Nominated [40]
Best Book of a Musical David Lindsay-Abaire Nominated
Best Original Score Jeanine Tesori (music) and David Lindsay-Abaire (lyrics) Nominated
Best Actor in a Musical Brian d'Arcy James Nominated
Best Actress in a Musical Sutton Foster Nominated
Best Featured Actor in a Musical Christopher Sieber Nominated
Best Orchestrations Danny Troob and John Clancy Nominated
Best Costume Design Tim Hatley Won
Drama Desk Award Outstanding Musical Nominated [41]
Outstanding Book of a Musical David Lindsay-Abaire Nominated
Outstanding Actor in a Musical Brian d'Arcy James Won
Daniel Breaker Nominated
Outstanding Actress in a Musical Sutton Foster Nominated
Outstanding Featured Actor in a Musical Christopher Sieber Nominated
Outstanding Director of a Musical Jason Moore Nominated
Outstanding Music Jeanine Tesori Nominated
Outstanding Lyrics David Lindsay-Abaire Nominated
Outstanding Orchestrations Danny Troob Nominated
Outstanding Set Design Tim Hatley Won
Outstanding Costume Design Won
Grammy Award Best Musical Show Album Nominated [42]
Original London production[edit]
Year
Award Ceremony
Category
Nominee
Result
Ref
2012 Laurence Olivier Award Best New Musical Nominated [43]
Best Actor in a Musical Nigel Lindsay Nominated
Best Performance in a Supporting Role in a Musical Nigel Harman Won
Best Costume Design Tim Hatley Nominated
Home media[edit]
In October 2009, Jeffrey Katzenberg said that a performance of the Broadway production had been recorded for a potential DVD release. However, due to the national tour and West End productions running considerably longer, the idea was put on-hold.[44] On July 19, 2013, following the closure of the national tour and West End productions, Amazon.com confirmed that the filmed performance would be available for instant viewing on September 17, 2013.[1] It also became available "in HD for playback on Kindle Fire HD, Xbox 360, PlayStation 3, Roku or other HD compatible devices" beginning October 15, 2013.[2] The home video release is also available on Netflix Streaming as of January, 2014.[45] A DVD, Blu-ray, and digital download was also released on that day.[3] The performance is an edit of several live performances as well as a performance shot without an audience. The original principal cast appear, as well as various alumni across the show's Broadway run. Also, it keeps the song "Donkey Pot Pie" instead of the replacement, "Forever."
References[edit]
^ Jump up to: a b SHREK: THE MUSICAL Now Available For Pre-Order Archived 2013-12-12 at the Wayback Machine., broadwayworld.com. Retrieved July 19, 2013.
^ Jump up to: a b Filmed Version of Broadway's Shrek Will Be Available in October Archived 2013-12-11 at the Wayback Machine. Playbill.com, Retrieved July 20, 2013
^ Jump up to: a b SHREK: THE MUSICAL DVD/Blu Now Available For Pre-Order Archived 2013-12-12 at the Wayback Machine., Broadway World, Retrieved July 26, 2013.
Jump up ^ Gurewitsch, Matthew. "Orchestrating an Ogre's Monster Makeover", The New York Times, December 11, 2008.
Jump up ^ Gans, Andrew."Keenan-Bolger and Sieber Are Part of Aug. 10 'Shrek' Reading" Archived 2008-12-06 at the Wayback Machine., Playbill.com, August 10, 2007.
^ Jump up to: a b Jacobson, Lynn. "Shrek the Musical", Variety, September 11, 2008.
Jump up ^ Gans, Andrew. "Seattle Goes Green: Shrek the Musical Makes World Premiere Aug. 14" Archived 2008-12-12 at the Wayback Machine., playbill.com, August 14, 2008.
Jump up ^ Jones, Kenneth. "Breaker Replaces Gregory and Other Changes Announced for NY Bound Shrek", playbill.com, September 30, 2008.
Jump up ^ Gans, Andrew."'Shrek the Musical' Adds "November Rain" to Broadway Score", playbill.com, September 30, 2009.
Jump up ^ Healy, Patrick. ‘" 'Shrek' to End Broadway Run on Jan. 3", New York Times, October 21, 2009.
Jump up ^ Gans, Andrew."Shrek Tour Will Launch in July 2010 in Chicago" Archived 2009-09-08 at the Wayback Machine., playbill.com, September 4, 2009.
Jump up ^ Gans, Andrew."Ogre There! National Tour of Shrek the Musical Kicks Off July 13 in Chicago" Archived 2010-07-16 at the Wayback Machine., playbill.com, July 13, 2010.
Jump up ^ "'Shrek the Musical' National tour Will Debut An All New Dragon", broadwayworld.com, September 12, 2009.
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^ Jump up to: a b "Tickets and Tour Schedule, NETworks Presentations", shrekthemusicalontour.com, accessed September 8, 2011.
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Jump up ^ "'Shrek the Musical' Begins Performances at West End's Theatre Royal, Drury Lane May 6", Playbill, retrieved August 10, 2014.
Jump up ^ Baz Bamigboye, "Ghost hopes to live again with its new musical", Mail Online, April 8, 2011.
Jump up ^ Bosanquet, Theo."Holden & Blackwood Confirmed for London 'Shrek'" Archived 2012-10-18 at the Wayback Machine., whatsonstage.com, July 30, 2010.
Jump up ^ "Nigel Lindsay and Nigel Harman cast in 'Shrek The Musical'", londontheatre.co.uk, 3 September 2010.
Jump up ^ SHREK to Undergo Changes for West End; New Characters & Design, broadwayworld.com.
Jump up ^ "Press reviews: 'Shrek the Musical'", BBC, 15 June 2011.
Jump up ^ News OfficialLondonTheatre.com
Jump up ^ News, whatsonstage.com.
Jump up ^ "Shrek closes on 24 Feb, Charlie moves Chocolate Factory to Drury Lane", whatsonstage.com.
Jump up ^ Shenton, Mark. " 'Shrek the Musical' to Embark on U.K./Ireland National Tour, Directed by Nigel Harman", playbill.com, 27 September 2013.
Jump up ^ Neil Cheesman, "Casting Announced for Shrek The Musical UK and Ireland Tour", londontheatre1.com, 3 February 2014.
Jump up ^ http://www.whatsonstage.com/london-theatre/news/amelia-lily-shrek-musical-cast_45048.html?cid=homepage_news
Jump up ^ Gans, Andrew. "Shrek CD Arrives in Stores March 24" Archived 2009-03-27 at the Wayback Machine., playbill.com, March 24, 2009.
Jump up ^ SHREK THE MUSICAL CD Debuts at #1 on Billboard's 'Top Cast Album' Chart.
Jump up ^ "The Billboard 200 'Shrek: The Musical'", billboard.com.
Jump up ^ BroadwayWorld.com OBCR News, broadwayworld.com June 10, 2009.
Jump up ^ Brantley, Ben. "The Belching Green Ogre Has a Song in His Heart", The New York Times, December 15, 2008.
Jump up ^ Rooney, David. "Shrek the Musical", Variety, December 14, 2008.
Jump up ^ Kuchwara, Michael, "A mean green ogre named Shrek moves to Broadway", Associated Press, December 14, 2008.
Jump up ^ Gardner, Elysa. "On Broadway, 'Shrek the Musical' is a gas, gas, gas", Usatoday.com, December 14, 2008.
Jump up ^ Gans, Andrew and Jones, Kenneth. "Nominations for 2009 Tony Awards Announced; Billy Elliot Earns 15 Nominations" Archived 2009-05-08 at the Wayback Machine., playbill.com, 5 May 2009.
Jump up ^ Gans, Andrew.Ruined and Billy Elliot Win Top Honors at Drama Desk Awards Archived 2009-05-21 at the Wayback Machine. playbill.com, 17 May 2009.
Jump up ^ "Ain't Misbehavin', Hair, 9 to 5, Shrek, West Side Story, Liza Nominated for GRAMMY Awards", broadwayworld.com.
Jump up ^ OLIVIERS 2012: All the Winners! MATILDA, ANNA CHRISTIE, Smith Pick Up Awards![permanent dead link]
Jump up ^ "'Shrek the Musical' headed for DVD release", Entertainment Weekly, Retrieved July 19, 2013.
Jump up ^ 'Sutton sparkles in Shrek the Musical on Netflix
External links[edit]
Production site for West End
Production site for Non-Equity Tour
Shrek the Musical at the Internet Broadway Database
Shrek the Musical at the Music Theatre International website
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Shrek
Films
Main films
Shrek (2001)
Shrek 2 (2004)
Shrek the Third (2007)
Shrek Forever After (2010)
Shrek 5 (TBA)
Spin-off films
Puss in Boots (2011)
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Shrek the Halls (2007)
Scared Shrekless (2010)
The Three Diablos (2012)
TV series
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Music
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