Tail expansions for random record distributions (Q2710567)

Let \(X_n\), \(n\in{\mathbf N}\), be independent random variables with distribution \(\mu\); the corresponding partial sums \(S_0=0\), \(S_n=\sum^n_{i=1} S_i\), constitute a renewal process with lifetime distribution \(\mu\). The random record distribution \(\nu\) associated with a probability distribution \(\mu\) can be written as a convolution series in \(\mu\), \(\nu=\sum^\infty_{n=1} {1\over n( n+1)} \cdot\mu^{*n}\), where * denotes the convolution so that \(\mu^{*n}\) is the distribution of \(S_n\). The traditional approaches in the investigation of the behaviour for the tails \(\nu((x,\infty))\), as \(x\to\infty\), are based on Laplace transforms and the associated Abel-Tauber type theorems.NEWLINENEWLINENEWLINEThe aim of the present paper is to show that the alternative approach of the so-called Banach algebra method can also be applied to random record distributions. For obtaining expansions of the tails under the moment conditions on \(\mu\), the Banach algebra method is using essentially the Gelfand transforms and the Wiener-Lévy-Gelfand theorem. Based on the Banach algebra method, the authors' results differ significantly from those obtained for other convolutions series that involve renewal measures, harmonic renewal measures, and compound distributions with exponentially decreasing weights.