Еxact rates in the Davis – Gut law of iterated logarithm for the first moment convergence of independent identically distributed random variables
Let $\{X, X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables and let $S_n = \sum^n_{i=1} X_i$, $M_n = \max_{1\leq k\leq n} |S_k|$. For $r > 0$, let $a_n(\varepsilon)$ be a function of $\varepsilon$ such that $a_n(\varepsilon ) \mathrm{l}\mathrm{o}\mathr...
Збережено в:
| Дата: | 2017 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2017
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/1689 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | Let $\{X, X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables and let $S_n = \sum^n_{i=1} X_i$,
$M_n = \max_{1\leq k\leq n} |S_k|$.
For $r > 0$, let $a_n(\varepsilon)$ be a function of $\varepsilon$ such that $a_n(\varepsilon ) \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} n \rightarrow \tau$ as $n \rightarrow \infty$ and $\varepsilon \searrow
\surd r$.
If $EX^2I\{|X| \geq t\} = o(\text{log}\text{log}t)^{-1})$ as $t \rightarrow \infty$ , then, by using the strong approximation, we show that
$$\lim_{\varepsilon \searrow \surd r} \frac 1{-\text{log}(\varepsilon^2 - r)} \sum ^{\infty}_{n=1}\frac{(\text{log} n)^{r-1}}{n^{3/2}}E \Bigl\{ M_n - (\varepsilon + a_n(\varepsilon ))\sigma \sqrt{2n \text{log log} n} \Bigr\}_{+} = \frac{2\sigma \varepsilon^{-2\tau \sqrt{r}}}{\sqrt{2\pi}r}$$
holds if and only if $EX = 0, EX^2 = \sigma^2$, and $EX = 0, EX^2 = \sigma^2$ та $EX^2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )^{r-1}(\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} | X| )^{-\frac 12} < \infty$. |
|---|