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

Let X
1
, X
2
, … be i.i.d. sequence and ƒ(x) be a regular variation function with index r. Let N(n) be a positive integer valued random variable and equation in probability, where η is a positive random variable. When some conditions are satisfied, the asymptotic distribution of M
ƒ(N(n))
is derived.