Theorems of large deviations in the approximation by the compound Poisson distribution (Q1415853)

A random variable \(Y\) is said to have the compound Poisson distribution if its characteristic function \(f(t)={E}e^{itY}\) is such that \[ \log f(t)= \lambda\sum_{m=1}^{N}\bigl(e^{itm}-1\bigr)p_m,\;\lambda>0,\;p_m\geq 0,\;m=1,\ldots,N,\;p_N>0,\;\sum_{m=1}^N p_m=1. \] \(Y\) can be expressed as the sum \(\xi_1+\cdots+\xi_\eta\) of a random number of i.i.d. random variables, where \({P}(\xi_1=m)=p_m\), \(m=1,\ldots,N\), \(\eta\) has the Poisson distribution with the parameter \(\lambda>0\). Let \(X\) be a random variable taking integer nonnegative values with \({E}X={E}Y\), \({E}X^s<\infty\) for any \(s>0\). The problem is to approximate \(X\) by \(Y\). In this paper probabilities of large deviations \({P}(X\geq x)\) are investigated using the cumulant method, after a definition of cumulants \(\widetilde {\widetilde{\Gamma}}_k(X)\), suitable for the case of the compound Poisson distribution, is given. The results of the paper extend previous investigations on large deviations in the approximation by Poisson law done by the authors.