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Miklós Schweitzer 2001 Problem 11

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February 11, 2017
Miklos Schweitzerprobabilityrandom variablesprobability and stats

Problem Statement

Let ξ(k1,k2),k1,k2∈N\xi_{(k_1, k_2)}, k_1, k_2 \in\mathbb N be random variables uniformly bounded. Let cl,l∈Nc_l, l\in\mathbb N be a positive real strictly increasing infinite sequence such that cl+1/clc_{l+1}/ c_l is bounded. Let dl=log⁡(cl+1/cl),l∈Nd_l=\log \left(c_{l+1}/c_l\right), l\in\mathbb N and suppose that Dn=∑l=1ndl↑∞D_n=\sum_{l=1}^n d_l\uparrow \infty when n→∞n\to\infty
Suppose there exist C>0C>0 and ε>0\varepsilon>0 such that ∣E{ξ(k1,k2)ξ(l1,l2)}∣≤C∏i=12{log⁡+log⁡+(cmax⁡{ki,li}cmin⁡{ki,li})}−(1+ε)\left| \mathbb E \left\{ \xi_{(k_1,k_2)}\xi_{(l_1,l_2)}\right\}\right| \leq C\prod_{i=1}^2 \left\{ \log_+\log_+\left( \frac{c_{\max\{ k_i, l_i\}}}{c_{\min\{ k_i, l_i\}}}\right)\right\}^{-(1+\varepsilon)} for each (k1,k2),(l1,l2)∈N2(k_1, k_2), (l_1,l_2)\in\mathbb N^2 (log⁡+\log_+ is the positive part of the natural logarithm). Show that lim⁡n1→∞n2→∞1Dn1Dn2∑k1=1n1∑k2=1n2dk1dk2ξ(k1,k2)=0\lim_{\substack{n_1\to\infty \\ n_2\to\infty}} \frac{1}{D_{n_1}D_{n_2}}\sum_{k_1=1}^{n_1} \sum_{k_2=1}^{n_2} d_{k_1}d_{k_2}\xi_{(k_1,k_2)}=0 almost surely.
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