Abstract:
Consider a process $Y_n$, $n\geq 1$, satisfying stochastic difference equation
$
Y_n = A_nY_{n-1} + B_n,\; n\geq 1, \; Y_0\geq 0, $ where $(A_n, B_n)$, $n\geq 1$ are i.i.d. pairs of non-negative random variables. It is known that the stationary processes of the given form have (under some conditions) two important properties: theirs stationary distribution have a power law tail and maximum $M_n=\max\{Y_1, \ldots, Y_n \}$ for $n\to\infty$ grows asymptotically as a maximum of $[\theta n]$
independent random variables with the same distribution. The work is devoted to study of two numerical characteristics: the tail index $\kappa$ and extremal index $\theta$.
The main objectives of this work are the following:
1) calculation of extremal index $\theta$ for the cases when it is expressed explicitly;
2) estimation of the index $\theta$ for the cases when it is not expressed explicitly;
3) proof of limit theorems for index $\theta$;
4) calculation ot indices $\kappa$ and $\theta$ in the multivariate case.
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