  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.
