Estimates of the rate of uniform convergence of sub-Gaussian random series (Q2755275)

The authors deal with the multiple random series NEWLINE\[NEWLINES(\vec{t})=\sum_{k_1=1}^{\infty}\dots\sum_{k_d=1}^{\infty} \varphi_{k_1,\dots, k_d}(\vec{t})\xi_{k_1,\dots, k_d},NEWLINE\]NEWLINE where \(\varphi_{k_1,\dots, k_d}(\vec{t})\) is a sequence of functions and \(\xi_{k_1,\dots, k_d}\) is a family of strictly sub-Gaussian random variables [see \textit{Yu. V. Kozachenko} and \textit{O. M. Moklyachuk}, Theory Probab. Math. Stat. 50, 89-98 (1995); translation from Teor. Jmovirn. Mat. Stat. 50, 87-96 (1994; Zbl 0861.60027)]. They found estimates for the probability \(P\{\|c(\vec{t}) (S(\vec{t})-S_{\vec{l}}^{\vec{n}}(\vec{t})) \|_C\geq\varepsilon\}\), where \(S_{\vec{l}}^{\vec{n}}(\vec{t})=\sum_{\vec{l}\leq\vec{k}\leq\vec{n}} \varphi_{\vec{k}}(\vec{t})\xi_{\vec{k}}\) is a partial sum of the series.