A Geometric(1/2) Distribution Associated with Record Breaking
Let $X_i,i=0,1,\ldots$ be a sequence of iid random variables whose distribution is continuous. Associated with this sequence is the sequence $(i,X_i),i=0,1,\ldots$. Let ${\cal R}_{n}$ denote the set of Pareto optimal elements of $\{ (i,X_i):i=0,\ldots,n\}.$ We refer to the elements of ${\cal R}_{n}$ as the current records at time $n,$ and we define $R_n=\vert {\cal R}_n\vert,$ the number of such records. Observe that $R_n$ has $\{1,\ldots,n+1\}$ as its support. When $(n,X_n)$ is realized, it is a Pareto optimal element of $\{ (i,X_i)~:~i=0,\ldots,n\}$ and ${\cal R}_{n} \backslash (n,X_n) \subset {\cal R}_{n-1}.$ Then we refer to those elements of ${\cal B}_n = {\cal R}_{n-1} \backslash {\cal R}_{n}$ as the records broken at time $n.$ Let $B_n= \vert {\cal B}_n \vert.$ We show that $$P[B_n = k] \rightarrow 1/2^{k+1} \mbox{ for } k=0,1,2,\ldots.$$