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/σ3√N 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.
From this we can conclude that ρ/σ3√N 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.