Abstract:
The paper studies the limit distributions of the maximum of sums
$\max_{1\le k\le n}\sum_{l=1}^k\xi_{n,l}$ for the triangular array $\xi_{n,k}$,
$k=1,\ldots,n$, $n=1,2,\ldots\,$, of independent identically distributed
random variables in a singular series in cases where
$a_n=E\xi_{n,k}\to
0$ and/or 1) $a_n\sqrt n\to\infty$, or 2) $a_n\sqrt n\to-\infty$,
or 3) $a_n\sqrt n\to 0$ as $n\to\infty$.
The direct proof that the analytic expressions for limit laws
coincide was previously obtained
by different authors and is given. Moreover,
for these transient cases the convergence of the sequence of distributions
of maximums to the limit laws is proved with the help of the characteristic
functions method.
Keywords:triangular array, maximum of sequential sums, limit distributions, method of characteristic functions.
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