Approximation by a binomial type law (Q1324855)

We define ``probabilities'' (measures) of a binomial type (b.t.) distribution by \[ p_{n,k} = {n! \over k! (n-k)!} p_ 0^{n-k} p^ k_ 1, \quad k=0,1, \dots, n,\;n=1,2,\dots,\;p_ 0 + p_ 1 \leq 1,\;p_ 0, p_ 1 \geq 0. \tag{1} \] The generating function is equal to \((p_ 0 + p_ 1z)^ n\). B. t. ``probabilities'' can serve as approximations for probabilities of sums of \(n\) independent identically distributed lattice random variables (r.v.'s). This is the problem investigated in the paper. Consider a r.v. \(X\) taking values \(k=0,1,\dots\), having the maximal step of distribution equal to 1 and corresponding probabilities \(p_ k\), \(0<p_ 1 \leq p_ 0\). For generality we also consider further the probabilities \(p_{n,k}\) of the sum \(X_ n\) of \(n\) independent copies of the r.v. \(X\), \(n=1,2 \dots\). Such sums often appear in applications.