V. Papathanasiou

On upper bounds for the variance of functions of random variables

The upper bounds for the variance of a function g of a random variable X obtained in Cacoullos (1982) (for short CP) are improved in the case [mu] = E(X) [not equal to] 0. A main feature of these bounds is that they involve the second moment of the derivative or the difference of g. A multivariate extension for functions of independent random variables is also given.

An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

For an absolutely continuous (integer-valued) r.v. X of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order k holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237–260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. X, expressions that seem to be known only in particular cases (for the Normal, see [Houdre and Kagan, J. Theoret. Probab. 8 (1995) 23–30]; see also [Houdre and Perez-Abreu, Ann. Probab. 23 (1995) 400–419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.

Some characterizations of distributions based on order statistics

Let X(1), X(2) be the order statistics of a sample of size two from an absolutely continuous random variable X. We obtain upper bounds for the covariance of X(1), X(2), which can be used to derive characterizations for specific distributions. A bound for the variance of the ith order statistic X(i) from a sample of size n is also obtained.

Lower variance bounds and a new proof of the central limit theorem

Lower variance bounds are derived for functions of a random vector X, thus extending previous results. Moreover, the w-function associated X is shown to characterize its distribution, and a special application shows the multivariate central limit theorem.

Variance inequalities for covariance kernels and applications to central limit theorems

A simple estimate for the error in the CLT, valid for a wide class of absolutely continuous r.v.s, is derived without Fourier techniques. This is achieved by using a simple convolution inequality for the variance of covariance kernels or w-functions in conjunction with bounds for the total variation distance. The results are extended to the multivariate case. Finally, a simple proof of the classical Darmois--Skitovich characterization of normality is obtained.

Variance bounds by a generalization of the Cauchy-Schwarz inequality☆

A refinement of the Cauchy-Schwarz inequality is suitably exploited to yield upper and lower bounds for the variance of a function of a continuous random variable.

Bounds for the variance of functions of random variables by orthogonal polynomials and Bhattacharya bounds

Upper and lower bounds for the variance of a function g of a random variable X are obtained by expanding g in a series of orthogonal polynomials associated with the distribution of X or by using the convergence of Bhattacharya bounds for exponential families of distribution.

Distance in Variation between Two Arbitrary Distributions via the Associated w-Functions

Variational inequalities are obtained for total variation distance between two arbitrary probability measures in terms of the corresponding w-functions. The results are extended to the distance between a distribution of a sum of dependent variables and an arbitrary distribution. Several applications are given.

Poisson approximation for a sum of dependent indicators: an alternative approach

The random variables X 1, X 2, …, X n are said to be totally negatively dependent (TND) if and only if the random variables X i and ∑ j≠i X j are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X 1, x 2, …, X n with P[X i = 1] = p i = 1 - P[X i = 0], an upper bound for the total variation distance between ∑ n i=1 X i and a Poisson random variable with mean λ ≥ ∑ n i=1 p i . An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Characterizations of multidimensional exponential families by Cacoullos-type inequalities

Let the distribution of a random vector X belong to the multidimensional exponential family. A Cacoullos-type, lower-bound, inequality for the variance of g(X) is given, which is shown to characterize the exponential family.