Unimodality of the distribution of record statistics

Let X₁,X₂,..., be a sequence of independent and identically distributed random variables with absolutely continuous distribution function F. For n ≥ 1, we denote the order statistics of X₁,X₂,...,X_n by X_1,n ≤ ... X_n,n. Define L(1) = 1, L(n+1) = min{j: j > L(n), X_j > X_j-1,j-1}, and X(n) = X_L(n),L(n), n ≥ 1. The sequence {X(n)} ({L(n)}) is called upper record statistics (times). In this article, we deal with unimodality of record statistics. We also deal with the strong unimodality of record statistics.