On the existence of exact upper sequences

B. A. Rogozin

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 4, pp. 701-707

Résumé

The following results are obtained. Theorem 2. Let $\xi_1,\xi_2,\dots,\xi_n,\dots$ be a sequence of independent random variables and $$ \frac{z^2\mathbf P\{|\xi_n-\mu(\xi_n)|>z\}}{\int_{|x|\le z}x^2\,d\mathbf P\{\xi_n-\mu(\xi_n)<x\}}\ge c>0, $$ $n=1,2,\dots$, then there exists no sequence $a_1,a_2,\dots,a_n,\dots$, $a_n\uparrow\infty$ as $n\to\infty$, having the property $$ \mathbf P\biggl\{\varlimsup_{m\to\infty}\frac{|S_n-\mu(S_n)|}{a_n}=1\biggr\}=1,\eqno(*) $$ where $S_n=\sum_{k=1}^n\xi_k$ and $\mu(\eta)$ is the median of $\eta$. Theorem 4. Let $\xi_1,\xi_2,\dots,\xi_n,\dots$ be a sequence of independent equally distributed random variables, then there exists no sequence $a_1,a_2,\dots,a_n,\dots$ with the properties $(*)$ and $$ \sum_{k=n}^\infty a_k^{-2}\le Cna_n^{-2} $$ for all $n$ and $C>0$. In the end of the paper an example is constructed which gives the negative answer to the question stated in [1].