Binomial Approximation to the Poisson Binomial Distribution: The Krawtchouk Expansion

By B. Roos. 2001

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A Proof of the Block Model Threshold Conjecture (2013, background): “Indeed, for each vertex u outside U we have that , It is well known (see e.g. [25]) that Next we note that with high probability | S ∗ | = (cid:100)√ loglog n (cid:101) since the probability that there exists a vertex in V (cid:48)(cid:48) with that number of neighbors tends to one; we will…”
Exact Kolmogorov and total variation distances between some familiar discrete distributions (2006, result): “Let n ∈ N∗, 0 < p < 1, and q := 1− p. Denote by Sn(p) a random variable having the binomial distribution with parameters n and p, that is, P ( Sn(p)= k ) := ( n k ) pkqn−k, k = 0,1, . . . ,n. (2.13) The well-known binomial-beta relation (cf. Johnson et al. [4, page 117 ]) reads as P ( Sn(p)≥ k )= n ”