Characterizations of distributions by variance bounds

Unified Variance Bounds and a Stein-Type Identity

“…The known identity of [4], (2.4) below, requiring an interval support of X, follows immediately from (2.2). Indeed, when S(X) is a (finite or infinite) interval, S(X) can always be taken to be open, and then S(X) = S(X * ).…”

“…An application of Tonelli's Theorem shows that f * integrates to 1, and therefore it is indeed a probability density (c.f. [4], [6] …”

“…Upper and lower variance bounds of g(X) for an arbitrary r.v. X were considered in [1] and [3] (see also [4]- [7] and references therein). Both upper and lower variance bounds may be obtained as by-products of the Cauchy-Schwarz inequality.…”

Binomial Approximation to the Generalized Hypergeometric Distribution

“…For the w-function associated with the generalized hypergeometric random variable, following [2], if E|w(X)∆g A (X)| < ∞ then…”

Negative Binomial Approximation to the Beta Binomial Distribution

“…The next relation is an important property to obtain the desired result, which was stated by [2]. If a non-negative integer-valued random variable X has p X (x) > 0 for every x in support of X and 0 < σ 2 = Var(X) < ∞, then…”