Asymptotic (r- 1)-dependent representation for rth order statistic from a stationary sequence

Adam Jakubowski

Stochastic Processes and their Applications, 1993, vol. 46, issue 1, 29-46

Abstract: Let X1, X2,... be a stationary sequence of random variables. Denote by M(k)n the kth largest value of X1, X2, ..., Xn. We find necessary and sufficient conditions for the existence of an (r- 1)-dependent stationary sequence X1,X2, ...(determined by a distribution function G and numbers [beta]1,[beta]2,...,[beta]r[greater-or-equal, slanted] 0,[Sigma]rq=1[beta]q=1), such that for each 67 as n-->+[infinity], where 32 are order statistics of 1. If such asymptotic (G, [beta]1, [beta]2, ...,[beta]r)-representation exists, then for each k, 1[less-than-or-equals, slant]k[less-than-or-equals, slant]r, there are numbers 0[less-than-or-equals, slant][gamma]k,j[less-than-or-equals, slant]1,j=1, 2, ..., k-1, satisfying 49 as n-->[infinity]. This corresponds to limit theorems for M(q)n obtained by Hsing (1988). Convergence of all order statistics is also discussed.

Date: 1993
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