On a Combinatorial Limit Theorem

Berry-Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing

“…Then, with C as in (22), conclusions (9) and (10) of Theorem 1.1 hold for the sum Y = n i=1 a i,π(i) with A = 8C/σ when A ≤ 1/12. Proof: Given π ′ , take (I, J) to be independent of π ′ , uniformly over all pairs with 1 ≤ I = J ≤ n, and set π ′′ = π ′ τ I,J .…”

“…In Theorem 2.5, below, where π is uniformly distributed over S n , this assumption is equivalent to (22). In Theorem 2.6, since π has no fixed points, by (27), without loss of generality we have a ii = 0 for all i in (26).…”

“…Our framework includes the case considered in [22], the uniform distribution over permutations with a single cycle. Consider a permutation π ∈ S n represented in cycle form; in S 7 for example, π = ((1, 3, 7, 5), (2,6,4)) is the permutation consisting of one 4 cycle in which 1 → 3 → 7 → 5 → 1 and one 3 cycle where 2 → 6 → 4 → 2.…”

“…The construction in Theorem 2.6 preserves the cycle structure in general and, when there are m 2-cycles, specializes to one similar, but not equivalent, to that of [18]. Theorem 2.6 With n ≥ 4, let an array {a ij } n ij=1 of real numbers satisfy (22), let…”

A Constant on a Uniform Bound of a Combinatorial Central Limit Theorem

“…The same condition was also shown to be necessary in the case of η n by Hajek (Hajek, J., 1961). In 1972, Robinson (Robinson, J., 1972) obtained necessary and sufficient conditions for the moments of η n to converge to those of a normal distribution and Kolchin and Chistyakov (Kolchin, V.F., 1973) considered a different η n where π is no longer uniform but attributes equal probabilities to only those permutations with one cycle.…”