Polynomial Time Computable Triangular Arrays For Almost Sure Convergence

Abstract

For 0 < x < 1, take the binary expansion with infinitely many 0's, replace each 0 with -1, this gives the polarized binary expansion of x. Let R_i(x) be the ith "polarized bit" and let S_n(x) be the sum of the first n R_i(x). {S_n} is the Z-valued random walk on (0,1). Normalize, by dividing each S_n by the square root of n: the resulting sequence converges weakly to the standard normal distribution on (0,1). The quantiles of S_n are random variables on (0,1), denoted S*_n, which are equal in distribution to the S_n, Skorokhod showed that the sequence of normalized quantiles converges almost surely to the standard normal distribution on (0,1). For n > 2, S*_n cannot be represented as the sum of the first n terms of a fixed sequence, R*_i, of random variables with the properties of the R_i. We introduce a method of constructing, for each n, an i.i.d family, R*_(n,1), ... R*_(n,n) which sums to S*_n, pointwise, as a function, not just in distribution. Each R*_(n,i) is a mean 0, variance 1 Rademacher random variable depending only on the first n bits. For each n, we get a bijection between the set of all such i.i.d. families, R*_(n,1), ... R*_(n,n), and the set of all admissible permutations of {0, ..., (2^n)-1}. Varying n, any doubly indexed such family, gives a triangular array representation of the sequence {S*_n} which is strong (because for each n, S*_n is the pointwise sum of the R*_(n,i)). Such representations are classified by sequences of admissible permutations. We show that the complexity of any sequence of admissible permutations is bounded below by that of 2^n. We explicitly construct three such polynomial time computable sequences whose complexity is bounded above by that of the function SBC (sum of binomial coefficients). We also initiate the study of some additional fine properties of admissible permutations.