On the closeness of the distributions of the two sums of independent random variables

V. M. Zolotarev

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 519-526

Résumé

Let $\{\xi_j\}$, $j=1,2,\dots,n$ (resp. $\{\eta_j\}$, $j=1,2,\dots,n$) be independent random variables with distribution functions $\{F_j\}$, $j=1,2,\dots,n$ (resp. $\{G_j\}$, $j=1,2,\dots,n$) and let $F$ (resp. $G$) be the distribution function of the sum $\xi=\xi_1+\dots+\xi_n$ (resp. $\eta=\eta_1+\dots+\eta_n$). Let us denote $$ \mu(k)=\sum_{j=1}^n\biggl|\int x^kd(F_j-G_j)\bigr|,\quad \nu(r)=\sum_{j=1}^n\int|x|^r|d(F_j-G_j)|. $$ We suppose that $\mu(0)=\mu(1)=\dots=\mu(m)=0$ and $\nu(r)$ exist for some $r$, $m\le r\le m+1$. In this case a) if the distribution of $\eta$ has a density bounded by a constant $q$, then $$ |F(x)-G(x)|[\nu(r)q^r]^\frac1{1+r},\eqno{(\text*)} $$ b) if $F$ and $G$ are lattice distributions with the same points of discontinuity and the same largest common factor of the length of the intervals between jumps $h$, then $$ |F(x)-G(x)|[\nu(r)h^{-r}]\eqno{(\text{**})} $$ where $C$ and $C_1$ are constants depending only on $m$ and $r$. In the case a) an estimation of the type (**), which is better then one of the type (*) can be achieved only when some additional requirements on $\xi_j$ are satisfied. The estimations (*) and (**) make it possible to formulate some sufficient conditions for $F$ to converge to infinitely divisible distribution $G$ when the summands $\xi_j$ are not necessarily uniformly infinitesimal.