## Limit distributions of extreme values of bounded independent random functions.(Ukrainian, English)Zbl 1098.60006

Teor. Jmovirn. Mat. Stat. 71, 114-122 (2004); translation in Theory Probab. Math. Stat. 71, 129-138 (2005).
For a sequence of independent Gaussian random functions the author studies the limit distribution of the probability that extreme values are located in the order interval. One of the obtained results is the following:
Let $$X=\{X(t),\;t\in T\}$$ be a bounded normally distributed random function, let $Z_{n}=\{Z_{n}(t)=\max\limits_{1\leq k\leq n}X_{k}(t),\;t\in T\};\quad R(t,s)=E(X(t)-EX(t))(X(s)-EX(s)),$
$\sigma(t)=(R(t,t))^{1/2},\quad GX=\{\sigma(t),\;t\in T\};$
$F_{\sup}(x)=P\left\{\sup\limits_{t\in T}X(t)<x\right\},\quad \Psi(x)=\Phi^{-1}(F_{\sup}(x)),$
where $$\Phi^{-1}(x)$$ is the inverse function to the standard Gaussian distribution function $$\Phi(x)$$; $d=\lim\limits_{x\to\infty}(\| GX\|\Psi(x)-x),\quad \theta(x)=\exp\{x^2/2\}(1-F_{\sup}(\| GX\| x-d)), \quad \| x\|=\sup_{t\in T}| x(t)|,$
$b_{n}=\begin{cases} (2\ln n)^{1/2}, & n>1;\\ 1, & n=1.\end{cases}$
If $$\| GX\|>0$$, then
$\lim\limits_{n\to\infty}P\left\{b_{n}\left({\| Z_{n}\|+d\over \| GX\|}-a_{n}\right)\leq x\right\}=\exp\{-e^{-x}\},$
where $$a_{n}=b_{n}+b_{n}^{-1}\ln(\theta(b_{n}))$$.

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60G70 Extreme value theory; extremal stochastic processes
Full Text: