Convergence of distributions of integral functionals of extremal random functions

A Limit Theorem for Integral Functionals of an Extremum of Independent Random Processes

“…However, in this case, we obtain only convergence in probability, in contrast to [9] where convergence in the mean of order k was established.…”

“…In [9], the moment conditions of [8] were considerably weakened. Furthermore, the general problem on conditions for the convergence of random variables f Y h n ( ) was considered in the case where finite-dimensional distributions of random processes Y t n ( ) converge to finite-dimensional distributions of a random process Y t ( ) that has independent values at every point.…”

“…Apparently for the first time, integral functionals of an extremum of independent random functions were studied in [8]. Under condition (6), which guarantees the asymptotic independence of extreme values of components of a process, it was established that degenerate laws are the limit ones.…”

“…Under condition (6), which guarantees the asymptotic independence of extreme values of components of a process, it was established that degenerate laws are the limit ones. Note that, in [8], the moment method was used, which resulted in the necessity of the existence of all moments of random functions.…”

“…In the proof of the main theorem on integral functionals of an extremum of independent random functions in [9], the following condition on the rate of decrease of distribution functions on the negative semiaxis was used among other conditions:…”