A characteristic martingale related to the counting process of records (Q2385603)

Let \((X_n)\) be a sequence of nonnegative, integrable, i.i.d. random variables with common distribution function \(F\). Let \(M_n=\max\{X_1,\dots,X_n\}\), \(n\geq 1\), \(M_0=-1\), be the process of partial maxima and let \(I_n=1_{\{X_n>M_{n-1}\}}\) be the indicator of an upper record. Consider the counting process of records \(N_n=\sum_{k=1}^n I_k,\) \({n\geq 1}\). The authors determine all distributions \(F\) such that \(N_n-cM_n\) is a martingale, for a given constant \(c>0\). Examples showing that the martingale property of \(N_n-cM_n\) characterizes the exponential and geometric distributions, among others, are given.