Limit theorems for record indicators in threshold $F^\alpha$-schemes

In Nevzorov's $F^\alpha$-scheme, one deals with a sequence of independent random variables with distribution functions that are powers of a common continuous distribution function. A key property of the $F^\alpha$-scheme is that the record indicators for such a sequence are independent. This allows one to obtain several important limit theorems for the total number of records in the sequence up to time $n\to\infty$. We extend these theorems to a much more general class of sequences of random variables obeying a "threshold $F^\alpha$-scheme" where the distribution functions of the variables are close to the powers of a common $F$ only in their right tails, above certain nonrandom nondecreasing threshold levels. Of independent interest is the characterization of the growth rate for extremal processes that we derive in order to verify the conditions of our main theorem. We also establish the asymptotic pairwise independence of record indicators in a special case of threshold $F^\alpha$-schemes.