On a loss of memory property of the maximum

Let {[xi]j} be a strictly stationary sequence which satisfies the condition D(un) for some sequence of constants {un}. Write Mn = max([xi]j: 1 [less-than-or-equals, slant] j [less-than-or-equals, slant] n). It is shown that if lim infn --> [infinity]P[Mn [less-than-or-equals, slant] un]>0, then limn --> [infinity]maxrn [less-than-or-equals, slant] q [less-than-or-equals, slant] n - rnnP[[xi]q+1>unMq [less-than-or-equals, slant] un] - P[Mrn>un]/rn= 0 where {rn} is a suitably chosen sequence of constants. This provides insight into some well-known results concerning the asymptotic properties of Mn.