On the limiting distribution of the maximum with random sample size of regular variation function

Let X₁, X₂, ... be i.i.d. sequence and f(x) be a regular variation function with index r. Let N(n) be a positive integer valued random variable and N(n)/n → η in probability, where η is a positive random variable. When some conditions are satisfied, the asymptotic distribution of Mf_(N(n)) is derived.