Precision of approximation of the generalized binomial distribution by convolutions of Poisson measures (Q1094739)

Let \(S_ n=X_ 1+X_ 2+...+X_ n\) where \(X_ 1,X_ 2,..\). are independent Bernoulli random variables with \(p(X_ j=1)=p_ j\) and \(p(X_ j=0)=1-p_ j\), \(j=1,2,...,n\). The author has studied the extents of precision of approximating \(F_ n(x)=P(S_ n<x)\) by distribution functions having Fourier-Stieltjes transforms of the types: \[ g_ 1(t)=\exp \{n\bar p(e^{it}-1)\},\quad \bar p=n^{- 1}\sum^{n}_{j=1}p_ j,\quad g_ 2(t)=\exp \{(n\bar p-\mu)(e^{it}- 1)+\mu it\}, \] \[ and\quad g_ 3(t)=\exp \{n\bar p(e^{it}-1)+\lambda (e^{2it}-1)\}, \] where \(\mu\) \(=\) greatest integer in \(\sum^{n}_{j=1}p^ 2_ j+\) and \(\lambda\)-\(\sum^{n}_{j=1}p^ 2_ j/2.\) Other similar estimates for \(F_ n(x+m)\), where m is any positive integer with \(m\leq n\), have also been considered. In each case, an upper bound of the errors of estimation, both local and integral, has been specified in terms of \(p_ j's\), \(j=1,2,...,n\); in some cases, lower bounds of such errors have been given. Interesting examples have also been discussed.