The property of extendability of limit distributions for the maximum term of a sequence

B. V. Gnedenko   ; L. Senusi-Bereksi

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (1983), pp. 11-20

Résumé

Let $\xi_1,\xi_2,\dots$ be a sequence of identically distributed independent random variables, and let $$ \eta_n=\max(\xi,\xi_2,\dots,\xi_n). $$ The following theorem is proved: If for a certain choice of constants $b_n>0$ and $a_n$ $$ P\biggl\{\frac1{b_n}(\eta_n-a_n)<x\biggr\}\to\Phi(x),\quad n\to\infty, $$ where $\Phi(x)$ is one of the three possible limiting distributions, and if the convergence is fulfilled in an interval $(c,d)$ for which $\Phi(d)-\Phi(c)>0$, then the convergence holds for all values of $x$. Библиогр. 5.