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Short Communications
On the closeness of the distributions of the two sums of independent random variables
V. M. Zolotarev Moscow
Abstract:
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
$$
|F(x)-G(x)|<C[\nu(r)q^r]^\frac1{1+r},\eqno{(\text*)}
$$
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
$$
|F(x)-G(x)|<C_1[\nu(r)h^{-r}]\eqno{(\text{**})}
$$
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.
Received: 10.05.1965