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Городской семинар по теории вероятностей и математической статистике
25 ноября 2011 г. 18:00, г. Санкт-Петербург, ПОМИ, ауд. 311 (наб. р. Фонтанки, 27)
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Теоремы сравнения для малых уклонений взвешенных сумм
Л. В. Розовский |
Количество просмотров: |
Эта страница: | 168 |
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Аннотация:
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. {\it 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 $\sum_{n\ge 1} |1-a_n/b_n|<\infty$,
then as $r\to 0^+$}
$$
\mathbb{P}\biggl(\sum_{n\ge 1} a_n\,X_n < r\biggr)\sim
\biggl(\prod_{n\ge 1} b_n/a_n\biggr)^\beta\,\mathbb{P}\biggl(\sum_{n\ge 1} b_n\,X_n < r\biggr).
$$
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