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Seminar on Probability Theory and Mathematical Statistics
November 25, 2011 18:00, St. Petersburg, PDMI, room 311 (nab. r. Fontanki, 27)
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Comparison theorems for small deviations of weighted series
L. V. Rozovskii |
Number of views: |
This page: | 168 |
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Abstract:
We study comparison theorems for small deviation probabilities of
weighted series and obtain more refined versions of the previous
results by the theme. In particular, we prove the following
result.
Theorem. Let a positive random variable $X$ belong to
the domain of attraction of a stable law with an index more than 1
and let its distribution function be regularly varying at zero
with an exponent $\beta>0$. If $\{X_n\}_{n\ge 1}$ are independent
copies of $ X$, and $\{a_n\}$ and $\{b_n\}$ are positive summable
sequences such that $ \sum\limits_{n\ge 1} |1-a_n/b_n|<\infty,$
then as $r\to 0^+$
$$
\mathbb{ P}\Big(\sum\limits_{n\ge 1} a_n\,X_n < r\Big)\sim
\Big(\prod\limits_{n\ge 1} b_n/a_n\Big)^\beta\,\mathbb{P}\Big(\sum\limits_{n\ge 1} b_n\,X_n < r\Big).
$$
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