Central Limit Theorems

In the previous chapter, we have studied some random variables which come up naturally when a stationary Poisson hyperplane process is observed inside some compact window. When the observation window is increased by a dilatation factor growing to infinity, the question for limit theorems satisfied by these random variables arises. The present chapter deals with central limit theorems for the standardized random variables. The first section exemplifies how a classical approach, employing U-statistics and characteristic functions, can be used. The next section explains how this approach can be refined to achieve a central limit theorem with explicit rate of convergence if the distance of the standardized random variables from a standard Gaussian random variable is measured in terms of the Wasserstein distance. The argument is based on a quantitative central limit theorem for random sums of independent and identically distributed random variables with an independent Poisson distributed number of summands. The latter result is derived in the final section by a combination of Stein’s method and coupling arguments.