Abstract:
Let $\overline S_n=\max_{1\le k\le n}\sum_{i=1}^{k}X_{i,n}$, where for any $n=1,2,\dots$ the
sequence $X_{1,n},\dots, X_{n,n}$ consists of independent and
identically distributed random
variables with finite positive variances. This paper studies the
problem of obtaining simple and
unimprovable sufficient conditions of the Lindeberg type which
guarantee the convergence of the
normalized variable $(\overline S_n-A_n)/B_n$ to a nondegenerate random
variable when the constants
$A_n$ and $B_n>0$ are chosen, respectively. The results
that Prokhorov and Borovkov
obtained are simplified, refined, and strengthened. In particular,
an unexplored case of when $D X_{1,n}\to 0$
as $n\to\infty$ is considered in detail.
Keywords:triangular array, maximum of sequential sums, uniform convergence of distributions, limit distributions, invariance principle, Prokhorov distance.
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