Record times

Abstract:

Let ${X_1},{X_2}, \cdots$ be independent and identically distributed random variables with a continuous distribution function, $L(n) \geq 2$ is called a record time if ${X_{L(n)}}$ is strictly larger than any previous ${X_j}$, and we put $L(1) = 1$. The sequence $1 = L(1) < L(2) < L(3) < \cdots$ of record times is a strictly increasing sequence of random variables. In the present note we investigate the sequence $\{ L(n)\}$ through the ratios $U(n) = L(n)/L(n - 1),n \geq 2$. We use an integer valued approximation $T(n)$ to $U(n)$, defined as the smallest integer such that $U(n) \leq T(n)$. These approximations turn out to be independent and identically distributed. This fact makes it possible to deduce several limit laws for $U(n)$ and for $\Delta (n) = L(n) - L(n - 1),n \geq 2$.