Convergence rate estimates in local limit theorems for Poisson random sums
Let \(g_{\lambda}\) be the density of absolutely continuous component of the distribution of \(S_{\lambda}/\sqrt{\lambda}\) where \(S_{\lambda} = \sum_{j=1}^{N_{\lambda}} X_j\). Here \(N_{\lambda}\) is a Poisson r.v. with parameter \(\lambda >0\); \(X_1,X_2,\ldots\) are i.i.d. r.v.'s with bounded density having \(EX_1=0\), \(EX_1^2 =1\) and \(E|X_1|^3 = \beta^3 < \infty\); \(N_{\lambda}\) and \(\{X_j\}\) are independent. If the ch.f. of \(X_1\) is integrable, then an explicit bound is given for \(\Delta(\lambda)= \sup_{x} |g_{\lambda} (x) - \varphi (x)|\) where \(\varphi\) is a density of a standard normal r.v. In particular, \(\limsup_{\lambda \to \infty} \sqrt{\lambda} \Delta (\lambda) \leq 0.2388 \beta^3\). Without the integrability assumption on the ch.f. of \(X_1\) analogues of the above mentioned results are obtained for appropriately smoothed densities \(g_{\lambda}\) and \(\varphi\).
scientific article; zbMATH DE number 1529959