A New Direct Proof of the Central Limit Theorem

Polynomial time relatively computable triangular arrays for almost sure convergence

“…We extend the methods and many (but not all) of the results of the latter to the setting of multinomial distributions provided by the former. These multinomial distributions essentially capture (via sufficiently good approximation) the full generality of situations where the Central Limit Theorem holds: this is one of the main results of [2]. Thus, the results of this paper represent significant progress on the program laid out in the first paragraph of [3]: obtaining (and analyzing the complexity of) strong trim triangular representations for the sequences of quantiles of weakly convergent sequences.…”

“…The two papers, [2] and [3], provide the context for our work here. The first of these papers provides a direct proof of the Central Limit Theorem that proceeds by introducing a sequence of approximating multinomial distributions.…”

“…We will say more, below, about the connections and differences between this paper, and the papers [2] and [3], but both of these end up working closely with a sequence, (R n |n ∈ Z + ), of i.i.d. random variables defined on (0, 1); both consider S n := n i=1 R n .…”

“…The main concern of [3] was, for a very specific choice of R n , to provide strong trim triangular array representations, R * n,i |n ∈ Z + and 1 ≤ i ≤ n of the sequence, (S * n |n ∈ Z + ) of quantiles of the sequence (S n |n ∈ Z + ), and to analyze the complexity of the simplest of these representations. We carry out this program here, starting from the more complicated and general sequences (R n |n ∈ Z + ) whose partial sums, S n , are the multinomial distributions of [2].…”

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We extend the methods and results of [3] to the setting of multinomial distributions satisfying certain properties. These include all the multinomial distributions arising from the direct proof of the Central Limit Theorem given in [2], which, by results of that paper, constitutes essentially full generality for the situations in which the Central Limit Theorem holds.

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