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Total variation asymptotics for sums of independent integer random variables. (English) Zbl 1018.60049

The authors derive an asymptotic expansion for the probability \(P[W_n\in A]\) on an arbitrary subset \(A\in \mathbb{Z}\), where \(W_n:=\sum^n_{j=1} Z_j\) is a sum of independent integer-valued random variables. After an Introduction, in Section 2 the authors establish properties of the solution of the Stein equation for certain signed measures on the integers. Section 3 deals with the simplest case, that of approximation by a centered Poisson distribution. For a large class of integer-valued random variables, the centered Poisson approximation already extends both the classical Poisson and the normal approximation. In Section 4 they move on to second-order expansions, concentrating on the case when the approximations are probability measures, as is relevant to Kolmogorov’s problem.
The main asymptotic expansion is proved in Section 5. Firstly it is supposed that the random variables \(Z_i\) are (integrably) “centered” in such a way that all the partial sums \(S_{rs} := \sum^s_{i=r}Z_i\) have second factorial cumulant satisfying \[ |k_2(S_{rs})|= |\text{Var} S_{rs}-ES_{rs}|\leq 1,\quad 1\leq r,s \leq n,\tag{1} \] as is clearly possible. Then, the following assumptions are made \[ \sigma^2_i :=\text{Var }Z_i\geq 2\quad\text{and}\quad 1 - d_{TV}(\ell(Z_i),\ell(Z_i+1))\geq \nu_*\tag{2} \] for all \(1\leq i\leq n\), where \(0 <\nu_*\leq 1/2\). The main result is given (Theorem 5): If \(Z_i\), \(1 \leq i\leq n\), are independent integer-valued random variables with finite \((r+1)\)th moments, which also satisfy (1) and (2), and if \(W = \sum^n_{i=1} Z_i\), then \(\|\ell(W)-\nu_r\|\leq 3Kn^{-r/2} (1+\tau_4)\), where \(K:=K (r, \tau_0, \tau_1,\tau_2,\tau_3,\nu_*) := \max \{C_1+C_2+C_3 + C_4, C_7,C_6(n_0)^{r/2}\}\). It is a very good and interesting paper.

MSC:

60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
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