Details for error bound on normal approximation to the Poisson distribution

This page is an appendix to the page Error in the normal approximation to the Poisson distribution.

The previous page noted that a Poisson random variable with mean λ has the same distribution as the sum of N independent Poisson random variables Xi each with mean λ/N. The Berry-Esséen theorem says that the error in approximating the CDF of ∑Xi with the CDF of its normal approximation is uniformly bounded by C ρ/σ3N where C is a constant we discuss below, ρ = E(|Xi − λ/N|3) and σ, the standard deviation of Xi, equals (λ/N)1/2.

Since the number N is arbitrary, we pick it as large as we like. This paper says the constant in the Berry-Esséen theorem can be made less than 0.7164 if N ≥ 65. This is no problem here because we will be taking the limit as N goes to infinity.

Next we calculate ρ, the third absolute moment of Xi. Let ω = λ/N. We can get an upper bound on ρ as follows.

\begin{eqnarray*}<br /> \rho \sum_{n=0}^\infty e^{-\omega} | n - \omega |^3 \frac{\omega^n}{n!} \\<br /> \sum_{n=0}^\infty e^{-\omega} (n + \omega)^3 \frac{\omega^n}{n!} \\<br /> \omega + 6\omega^2 + 8\omega^3.<br /> \end{eqnarray*}

From this we can conclude that ρ/σ3N equals λ plus term involving 1/N. As we let N to infinity, the latter terms drop out and so we have the error bound on the normal approximation to the Poisson of C/√λ where C < 0.7164.