Properties of record values with random index characterizing exponential distribution (Q2770146)

Let \((X_n\), \(n\geq 1)\) be a sequence of independent, non-negative, absolutely continuous and identically distributed random variables. Consider the sequence \((R_\nu)\) of its record values indexed by a geometrically distributed random variable \(\nu\) (with parameter \(p\)). Under some conditions, the random variable \(X_1\) is exponentially distributed if and only if the random variables \(pR_\nu\) and \(R_{\nu+1}-R_\nu\) are identically distributed. The same is valid for the expected values and failure rates.