On the second record derivative of a sequence of exponential random variables

Abstract

Let Zi (i ≥ 1) be a sequence of independent and identically distributed random variables with standard exponential distribution H and Z(n) (n ≥ 1) be the corresponding sequence of exponential records associated with Zi (i ≥ 1). Let us call the sequence Z(n) (n ≥ 1) the first “record derivative” of the sequence Zi (i ≥ 1). It is known that ν1 = Z(1), ν2 = Z(2) − Z(1), . . . are independent variables with distribution H. Let T (n) (n ≥ 1) be record times obtained from the sequence ν1, ν2, . . . and Y (n) = Z(T (n)),W(n) = Y (n) − Y (n − 1) (n ≥ 1). Let us call the sequence Y (n) (n ≥ 1) (the main objective of the research of the present paper) the second “record derivative” of the sequence Zi (i ≥ 1). In the present paper, we find the distributions of T (n), Y (n), W(n) and study the Laplace transform of Y (n). A limit result for the sequence Y (n) (n ≥ 1) is obtained in the paper. We also propose some methods of generation of T (n) and Y (n).

Keywords:

record values, exponential distribution, limit results, methods of record generation