A Geometric(1/2) Distribution Associated with Record Breaking

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