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[greater-or-equal, slanted]1, we denote the order statistics of X1,X2,...,Xn by X1,n[less-than-or-equals, slant]X2,n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn,n. Define L(1)=1, L(n+1)=min{j: j>L(n), Xj>Xj-1,j-1}, andX(n)=XL(n),L(n), n[greater-or-equal, slanted]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.