The Spacings of Record Values

Let {Ri; i ≥ 1} be the sequence of record values generated from a sequence of i.i.d. continuous random variables, coming from a big class of distributions in the exponential family. In this paper we study the behavior of k-spacings of Ri, that is, Rk+i−Ri for i ≥ 1. We show that, under certain conditions, a normalized k-spacings empirically converges to a Gamma distribution. Counter examples show that the result is not valid when the conditions are violated. A strong law and a limiting distribution of the largest normalized spacings are also derived. In particular, these results conclude that the k-spacings go to infinity when the population distribution has heavy tail; the spacings go to zero when the tail is not heavy. Exact speeds of such convergence are obtained. Supported in part by NSF #DMS-0308151 and NSF #DMS-0449365, School of Statistics, University of Minnesota, 224 Church Street, MN55455, [email protected]. School of Mathematics, Jilin University, 10 Qianwei road, Changchun130012, P. R. China, [email protected].