Theorem

Renyi

Assume $\{X_n, n \geq 1\}$ are iid with common, continuous distribution function $F(x)$. We say $X_n$ is a record of the sequence if $X_n > \bigvee_{i=1}^{n-1} X_i$, and define $A_n = [X_n \text{ is a record}]$. We also define $R_n = \sum_{j=1}^{n} \mathbb{1}_{[X_j \geq X_n]}$. Then

1. the sequence of random variables $\{R_n, n \geq 1\}$ is independent and $\mathbb{P}[R_n = k] = \frac{1}{n}$ for $k = 1, \dots , n$;
2. the sequence of events $\{A_n, n \geq 1\}$ is independent and $\mathbb{P}(A_n) = \frac{1}{n}$.