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).