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

Let $Z_i (i \geqslant 1)$ be a sequence of independent and identically distributed random variables with standard exponential distribution $H$ and $Z(n)(n \geqslant 1)$ be the corresponding sequence of exponential records associated with $Z_i(i \geqslant 1)$. Let us call the sequence $Z(n)(n \geqslant 1)$ the first "record derivative" of the sequence $Z_i(i \geqslant 1)$. It is known that $\nu_1 = Z(1), \nu2 = Z(2) - Z(1)$, . . . are independent variables with distribution $H$. Let $T(n)(n \geqslant 1)$ be record times obtained from the sequence $\nu_1, \nu_2, \ldots $ and $Y(n) = Z(T (n)), W(n) = Y (n) - Y(n - 1) (n \geqslant 1)$. Let us call the sequence $Y(n) (n \geqslant 1)$ (the main objective of the research of the present paper) the second "record derivative" of the sequence $Z_i(i \geqslant 1)$. 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 $Y(n)(n \geqslant 1)$ is obtained in the paper. We also propose some methods of generation of $T(n)$ and $Y(n)$.