An additive property of weak records from geometric distributions

Let ${W_m}{_{mge 1}}$ be the sequence of weak records from a discrete parent random variable, $X$ , supported on the non-negative integers. We obtain a new characterization of geometric distributions based on an additive property of weak records: $X$ follows a geometric distribution if and only if for certain integers, $n,, sge 1, W_{n+s}stackrel{d}{=}W_n+W^{prime }_s$ , with $W^{prime }_s$ independent of $W_n$ and $W^{prime }_sstackrel{d}{=} W_s$ .