Background: Maestro Bogomolny was interested in a probabilistic interpretation of the above, having noticed an earlier derivation that relied on the behavior of Poisson sums under the law of large numbers. For in a forcoming paper, the following was derived “probabilistically”:
From the behavior of the sum of Poisson variables as they converge to a Gaussian by the central limit theorem: where is a Poisson random variable with parameter . Since the sum of independent Poisson random variables with parameter is Poisson with parameter , the Central Limit Theorem says the probability distribution of approaches a standard normal distribution. Thus as .