New characterization of discrete distributions through weak records

Let $X_{1},X_{2},\ldots$ be a sequence of independent and identically distributed random variables taking on values $0,1,\ldots$ with a distribution function $F$ such that $F(n) < 1$ for any $n=0,1,\ldots$ and $\mathbf{E} X_{1}\log (1+X_{1}) < \infty $. Let $X_{L(n)}$ be the $n$th weak record value and $\{ A_{k}\}_{k=0}^{\infty }$ be any sequence of positive numbers, such that $A_{k+1} > A_{k}-1$. This paper shows that if there exists an $F(x)$, with $\mathbf{E} \{X_{L(n+2)}-X_{L(n)}\mid X_{L(n)}=s\}=A_{s}$ for some $n > 0$ and all $s\ge 0$, then $F(x)$ is unique.