A strong limit theorem under no assumption of independence, stationarity or various dependences

Let {qn, n [greater-or-equal, slanted] 0} be a sequence of positive integers, In = {0, 1, ..., qn}, {Xn, n [greater-or-equal, slanted] 0} a sequence of random variables, where Xn takes on values in In, and P(X1 = x1, ..., Xn = xn) > 0, for all xi [epsilon] Ii, 0 [less-than-or-equals, slant] i [less-than-or-equals, slant] n. The purpose of this paper is to give a strong limit theorem for the above sequence of random variables concerning conditional expectation without any assumption of independence, stationarity or various dependences.