Stability of a characterization of the exponential distribution
Let X be a positive random variable with a distribution function F(x) satisfying the inequality \(F(x)<1\). Suppose that, for some integer \(k>0\), X has a finite kth order moment; let \(EX^ k=a\). Further, suppose that, for all \(x\geq 0\), the inequality \[ (*)\quad | E\{X-x)^ k| X\geq x\}-a| (1-F(x))\leq \epsilon \] holds. Then there exist positive constants c and \(\epsilon_ 0\) (depending on k) such that, for all \(x\geq 0\) and \(\epsilon \in [0,\epsilon_ 0]\), the estimate \[ | F(x)-(1- e^{-\lambda x})| \leq c\epsilon^{\delta_ k} \] is valid, where \(0<\delta_ k\leq 1\) and \(\lambda =(k!/a)^{1/k}\). In particular, \(\delta_ k=1\) for \(k\leq 3\) and \(\delta_ 4\to 1\) as \(\epsilon\) \(\to 0\). For the case \(k>4\) the authors construct an example of a distribution function \(F_ 1(x)\) for which (*) is satisfied and the estimate \[ \sup_{x\geq 0}| F_ 1(x)-(1-e^{-\lambda x})| \geq c_ 1\epsilon^{\alpha} \] is valid, where \(c_ 1>0\) and \(0<\alpha <1\) are some constants.
scientific article; zbMATH DE number 3940402