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For 0 < x < 1, take the binary expansion with infinitely many 0's, replaceeach 0 with -1, this gives the polarized binary expansion of x. Let R_i(x) bethe 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 thesquare root of n: the resulting sequence converges weakly to the standardnormal 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, Skorokhodshowed that the sequence of normalized quantiles converges almost surely to thestandard normal distribution on (0,1).  For n > 2, S*_n cannot be represented as the sum of the first n terms of afixed sequence, R*_i, of random variables with the properties of the R_i. Weintroduce 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 indistribution. Each R*_(n,i) is a mean 0, variance 1 Rademacher random variabledepending only on the first n bits. For each n, we get a bijection between theset of all such i.i.d. families, R*_(n,1), ... R*_(n,n), and the set of alladmissible permutations of {0, ..., (2^n)-1}.  Varying n, any doubly indexed such family, gives a triangular arrayrepresentation of the sequence {S*_n} which is strong (because for each n, S*_nis the pointwise sum of the R*_(n,i)). Such representations are classified bysequences of admissible permutations. We show that the complexity of anysequence of admissible permutations is bounded below by that of 2^n. Weexplicitly construct three such polynomial time computable sequences whosecomplexity is bounded above by that of the function SBC (sum of binomialcoefficients). We also initiate the study of some additional fine properties ofadmissible permutations.

Polynomial Time Computable Triangular Arrays For Almost Sure Convergence