Central limit theorem for the number of records in Fα-scheme

Abstract

Consider the sequence {Xk, k ≥ 1} of independent identically distributed random variables whose distribution function is continuous. Then events of the type {Xi = Xj} have probability 0 if i ≠ j. Let L(1) = 1. For n ≥ 2, we define random variablesL(n) = inf{k > L(n − 1) : Xk > XL(n-1)}assuming that inf ∅ := +∞. The members of the sequence L = {L(n), n ≥ 1} are called moments of records constructed for {Xk, k ≥ 1}. Consider the sequence of random variables μ = {μ(n), n ≥ 1}, defined by the relationμ(n) = #{k : L(k) ≤ n}, n ≥ 1.It is clear that μ(n) – is the number of records that happened up to the moment n inclusive.In the work [10], the so-called Fα-scheme is considered for the first time, which is built using a given distribution function and a sequence of positive numbers {αk}. It is clear that Fαn(x) is the distribution function for each n ≥ 1. The set of independent random variables {Xn} is called the Fα scheme, if the distribution function of the random variable Xn is Fαn(x). If all αn are equal to each other, then the Fα scheme – is a set independent identically distributed random variables. If not all αn are equal to each other, then the Fα scheme – is a generalization of the classical case.This paper examines the assertions related to the fulfillment of the central limit theorem (CLT) for the number of records in the Fα-scheme of records. The method of finding exact asymptotic expressions for mathematical expectation and variance, which can be used to replace the real characteristics in CLT, is given.A specific example of power-law growth of exponents of the Fα-scheme was considered, and CLT is constructed only in terms of the moment of observation and the power of growth.The article contains 4 theorems with complete proof. Theorem 1 relates the mathematical expectation and variance to the accumulated intensity of the Fα-scheme. Theorem 2 establishes the implementation of CLT in general, and theorem 4 – for a specific case.