Poisson approximation via the convolution with Kornya–Presman signed measures

We present an upper bound for the total variation distance between the generalized polynomial distribution and a finite signed measure, which is the convolution of two finite signed measures, one of which is of Kornya–Presman type. In the one-dimensional Poisson case, such a finite signed measure was first considered by K. Borovkov and D. Pfeifer [J. Appl. Probab., 33 (1996), pp. 146–155].
We give asymptotic relations in the one-dimensional case, and, as an example, the independent identically distributed record model is investigated.
It turns out that here the approximation is of order $O(n^{-s}(\ln n)^{-{(s+1)/2}})$ for $s$ being a fixed positive integer, whereas in the approximation with simple Kornya–Presman signed measures, we only have the rate $O((\ln n)^{-(s+1)/2})$.