An additive property of weak records from geometric distributions

Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\{W_m\}{_{m\ge 1}}$$</EquationSource> </InlineEquation> be the sequence of weak records from a discrete parent random variable, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$X$$</EquationSource> </InlineEquation>, supported on the non-negative integers. We obtain a new characterization of geometric distributions based on an additive property of weak records: <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X$$</EquationSource> </InlineEquation> follows a geometric distribution if and only if for certain integers, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$n,\, s\ge 1, W_{n+s}\stackrel{d}{=}W_n+W^{\prime }_s$$</EquationSource> </InlineEquation>, with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$W^{\prime }_s$$</EquationSource> </InlineEquation> independent of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$W_n$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$W^{\prime }_s\stackrel{d}{=} W_s$$</EquationSource> </InlineEquation>. Copyright Springer-Verlag 2013