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Berry–Esseen theorem

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Theorem in probability theory

In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity. Under stronger assumptions, the Berry–Esseen theorem, or Berry–Esseen inequality, gives a more quantitative result, because it also specifies the rate at which this convergence takes place by giving a bound on the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean. The approximation is measured by the Kolmogorov–Smirnov distance. In the case of independent samples, the convergence rate is n, where n is the sample size, and the constant is estimated in terms of the third absolute normalized moment.

Statement of the theorem

Statements of the theorem vary, as it was independently discovered by two mathematicians, Andrew C. Berry (in 1941) and Carl-Gustav Esseen (1942), who then, along with other authors, refined it repeatedly over subsequent decades.

Identically distributed summands

One version, sacrificing generality somewhat for the sake of clarity, is the following:

There exists a positive constant C such that if X1, X2, ..., are i.i.d. random variables with E(X1) = 0, E(X1) = σ > 0, and E(|X1|) = ρ < ∞, and if we define
Y n = X 1 + X 2 + ⋯ + X n n {\displaystyle Y_{n}={X_{1}+X_{2}+\cdots +X_{n} \over n}}
the sample mean, with Fn the cumulative distribution function of
Y n n σ , {\displaystyle {Y_{n}{\sqrt {n}} \over {\sigma }},}
and Φ the cumulative distribution function of the standard normal distribution, then for all x and n,
| F n ( x ) − Φ ( x ) | ≤ C ρ σ 3 n .         ( 1 ) {\displaystyle \left|F_{n}(x)-\Phi (x)\right|\leq {C\rho \over \sigma ^{3}{\sqrt {n}}}.\ \ \ \ (1)}
Illustration of the difference in cumulative distribution functions alluded to in the theorem.

That is: given a sequence of independent and identically distributed random variables, each having mean zero and positive variance, if additionally the third absolute moment is finite, then the cumulative distribution functions of the standardized sample mean and the standard normal distribution differ (vertically, on a graph) by no more than the specified amount. Note that the approximation error for all n (and hence the limiting rate of convergence for indefinite n sufficiently large) is bounded by the order of n.

Calculated upper bounds on the constant C have decreased markedly over the years, from the original value of 7.59 by Esseen in 1942. The estimate C < 0.4748 follows from the inequality

sup x ∈ R | F n ( x ) − Φ ( x ) | ≤ 0.33554 ( ρ + 0.415 σ 3 ) σ 3 n , {\displaystyle \sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq {0.33554(\rho +0.415\sigma ^{3}) \over \sigma ^{3}{\sqrt {n}}},}

since σ ≤ ρ and 0.33554 · 1.415 < 0.4748. However, if ρ ≥ 1.286σ, then the estimate

sup x ∈ R | F n ( x ) − Φ ( x ) | ≤ 0.3328 ( ρ + 0.429 σ 3 ) σ 3 n , {\displaystyle \sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq {0.3328(\rho +0.429\sigma ^{3}) \over \sigma ^{3}{\sqrt {n}}},}

is even tighter.

Esseen (1956) proved that the constant also satisfies the lower bound

C ≥ 10 + 3 6 2 π ≈ 0.40973 ≈ 1 2 π + 0.01079. {\displaystyle C\geq {\frac {{\sqrt {10}}+3}{6{\sqrt {2\pi }}}}\approx 0.40973\approx {\frac {1}{\sqrt {2\pi }}}+0.01079.}

Non-identically distributed summands

Let X1, X2, ..., be independent random variables with E(Xi) = 0, E(Xi) = σi > 0, and E(|Xi|) = ρi < ∞. Also, let
S n = X 1 + X 2 + ⋯ + X n σ 1 2 + σ 2 2 + ⋯ + σ n 2 {\displaystyle S_{n}={X_{1}+X_{2}+\cdots +X_{n} \over {\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}+\cdots +\sigma _{n}^{2}}}}}
be the normalized n-th partial sum. Denote Fn the cdf of Sn, and Φ the cdf of the standard normal distribution. For the sake of convenience denote
σ → = ( σ 1 , … , σ n ) ,   ρ → = ( ρ 1 , … , ρ n ) . {\displaystyle {\vec {\sigma }}=(\sigma _{1},\ldots ,\sigma _{n}),\ {\vec {\rho }}=(\rho _{1},\ldots ,\rho _{n}).}
In 1941, Andrew C. Berry proved that for all n there exists an absolute constant C1 such that
sup x ∈ R | F n ( x ) − Φ ( x ) | ≤ C 1 ⋅ ψ 1 ,         ( 2 ) {\displaystyle \sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq C_{1}\cdot \psi _{1},\ \ \ \ (2)}
where
ψ 1 = ψ 1 ( σ → , ρ → ) = ( ∑ i = 1 n σ i 2 ) − 1 / 2 ⋅ max 1 ≤ i ≤ n ρ i σ i 2 . {\displaystyle \psi _{1}=\psi _{1}{\big (}{\vec {\sigma }},{\vec {\rho }}{\big )}={\Big (}{\textstyle \sum \limits _{i=1}^{n}\sigma _{i}^{2}}{\Big )}^{-1/2}\cdot \max _{1\leq i\leq n}{\frac {\rho _{i}}{\sigma _{i}^{2}}}.}
Independently, in 1942, Carl-Gustav Esseen proved that for all n there exists an absolute constant C0 such that
sup x ∈ R | F n ( x ) − Φ ( x ) | ≤ C 0 ⋅ ψ 0 ,         ( 3 ) {\displaystyle \sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq C_{0}\cdot \psi _{0},\ \ \ \ (3)}
where
ψ 0 = ψ 0 ( σ → , ρ → ) = ( ∑ i = 1 n σ i 2 ) − 3 / 2 ⋅ ∑ i = 1 n ρ i . {\displaystyle \psi _{0}=\psi _{0}{\big (}{\vec {\sigma }},{\vec {\rho }}{\big )}={\Big (}{\textstyle \sum \limits _{i=1}^{n}\sigma _{i}^{2}}{\Big )}^{-3/2}\cdot \sum \limits _{i=1}^{n}\rho _{i}.}

It is easy to make sure that ψ0≤ψ1. Due to this circumstance inequality (3) is conventionally called the Berry–Esseen inequality, and the quantity ψ0 is called the Lyapunov fraction of the third order. Moreover, in the case where the summands X1, ..., Xn have identical distributions

ψ 0 = ψ 1 = ρ 1 σ 1 3 n , {\displaystyle \psi _{0}=\psi _{1}={\frac {\rho _{1}}{\sigma _{1}^{3}{\sqrt {n}}}},}

and thus the bounds stated by inequalities (1), (2) and (3) coincide apart from the constant.

Regarding C0, obviously, the lower bound established by Esseen (1956) remains valid:

C 0 ≥ 10 + 3 6 2 π = 0.4097 … . {\displaystyle C_{0}\geq {\frac {{\sqrt {10}}+3}{6{\sqrt {2\pi }}}}=0.4097\ldots .}

The lower bound is exactly reached only for certain Bernoulli distributions (see Esseen (1956) for their explicit expressions).

The upper bounds for C0 were subsequently lowered from Esseen's original estimate 7.59 to 0.5600.

Sum of a random number of random variables

Berry–Esseen theorems exist for the sum of a random number of random variables. The following is Theorem 1 from Korolev (1989), substituting in the constants from Remark 3. It is only a portion of the results that they established:

Let { X i } {\displaystyle \{X_{i}\}} be independent, identically distributed random variables with E ( X i ) = μ {\displaystyle E(X_{i})=\mu } , Var ( X i ) = σ 2 {\displaystyle {\text{Var}}(X_{i})=\sigma ^{2}} , E | X i − μ | 3 = κ 3 {\displaystyle E|X_{i}-\mu |^{3}=\kappa ^{3}} . Let N {\displaystyle N} be a non-negative integer-valued random variable, independent from { X i } {\displaystyle \{X_{i}\}} . Let S N = X 1 + ⋯ + X N {\displaystyle S_{N}=X_{1}+\cdots +X_{N}} , and define
Δ = sup x | P ( S N − E ( S N ) Var ( S N ) ≤ z ) − Φ ( z ) | {\displaystyle \Delta =\sup _{x}\left|P\left({\frac {S_{N}-E(S_{N})}{\sqrt {{\text{Var}}(S_{N})}}}\leq z\right)-\Phi (z)\right|}
Then
Δ ≤ 3.8696 κ 3 E ( N ) σ 3 + 1.0395 E | N − E ( N ) | E ( N ) + 0.2420 μ 2 Var ( N ) σ 2 E ( N ) {\displaystyle \Delta \leq 3.8696{\frac {\kappa ^{3}}{{\sqrt {E(N)}}\sigma ^{3}}}+1.0395{\frac {E|N-E(N)|}{E(N)}}+0.2420{\frac {\mu ^{2}{\text{Var}}(N)}{\sigma ^{2}E(N)}}}

Multidimensional version

As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.

Let X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} be independent R d {\displaystyle \mathbb {R} ^{d}} -valued random vectors each having mean zero. Write S n = ∑ i = 1 n X i {\displaystyle S_{n}=\sum _{i=1}^{n}X_{i}} and assume Σ n = Cov ⁡ [ S n ] {\displaystyle \Sigma _{n}=\operatorname {Cov} } is invertible. Let Z n ∼ N ⁡ ( 0 , Σ n ) {\displaystyle Z_{n}\sim \operatorname {N} (0,{\Sigma _{n}})} be a d {\displaystyle d} -dimensional Gaussian with the same mean and covariance matrix as S n {\displaystyle S_{n}} . Then for all convex sets U ⊆ R d {\displaystyle U\subseteq \mathbb {R} ^{d}} ,
| Pr [ S n ∈ U ] − Pr [ Z n ∈ U ] | ≤ C d 1 / 4 γ n {\displaystyle {\big |}\Pr-\Pr\,{\big |}\leq Cd^{1/4}\gamma _{n}} ,
where C {\displaystyle C} is a universal constant and γ n = ∑ i = 1 n E ⁡ [ ‖ Σ n − 1 / 2 X i ‖ 2 3 ] {\displaystyle \gamma _{n}=\sum _{i=1}^{n}\operatorname {E} {\big }} (the third power of the L norm).

The dependency on d 1 / 4 {\displaystyle d^{1/4}} is conjectured to be optimal, but might not be.

See also

Notes

  1. Since the random variables are identically distributed, X2, X3, ... all have the same moments as X1.

References

  1. Esseen (1942). For improvements see van Beek (1972), Shiganov (1986), Shevtsova (2007), Shevtsova (2008), Tyurin (2009), Korolev & Shevtsova (2010a), Tyurin (2010). The detailed review can be found in the papers Korolev & Shevtsova (2010a) and Korolev & Shevtsova (2010b).
  2. Shevtsova (2011).
  3. Esseen (1942); Zolotarev (1967); van Beek (1972); Shiganov (1986); Tyurin (2009); Tyurin (2010); Shevtsova (2010).
  4. Korolev, V. Yu (1989). "Theory of Probability & Its Applications". Theory of Probability & Its Applications. 33 (3): 540–544. doi:10.1137/1133079.
  5. Bentkus, Vidmantas. "A Lyapunov-type bound in R." Theory of Probability & Its Applications 49.2 (2005): 311–323.
  6. ^ Raič, Martin (2019). "A multivariate Berry--Esseen theorem with explicit constants". Bernoulli. 25 (4A): 2824–2853. arXiv:1802.06475. doi:10.3150/18-BEJ1072. ISSN 1350-7265. S2CID 119607520.

Bibliography

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