On the relative stability of extremal random functions (Q1810128)

Let \(X=\{X(t),t\in T\}\) be a bounded normally distributed random function given on parametric set \(T\) and taking values in \(\mathbb{R}\). Put \(\sigma_T X=\{(E|X(t)-EX(t) |^2)^{1/2}\), \(t\in T\}\). For a sequence \((X_n)\) of independent copies of \(X\), define the extremal random functions \(Z_n=\{\sup_{1\leq k\leq n}X_k (t),t\in T\}\). The author proves that the sequence \((Z_n)\) is relatively stable almost everywhere, i.e. \(\lim_{n\to\infty} \|Z_n/b_n- \sigma_TX\|= 0\) almost everywhere, where \[ b_n=\begin{cases} \sqrt{2\ln n},\quad & n<1,\\ 1,\quad & n=1,\end{cases} \] and \(|x|= \sup_{t\in T}|x(t)|\) is a uniform norm. Reviewer's remark: The translation of the paper is careless.