Inequalities for the Concentration Functions of Sums of Independent Random Variables

Abstract

1. The concentration function of a one-dimensional random variable X is defined to be ifj(x) = xjj{x: X) = supPr(a < X ^ a + x) where the supremum is taken over all real values of a. In & dimensions we can define the concentration function ifj(x) = tfj(x 1} x 2 , ...,x k ) in a similar way, taking the supremum over all ^-dimensional intervals whose sides are parallel to the coordinate axes. We can also define a 'spherical' concentration functionwhere this time the supremum is taken over all x 0 e R k and |. | is the usual Euclidean distance in R k .In this paper, we look for upper bounds for the concentration functions of sums of independent random variables, both in R x and in R k (k > 1). Many such inequalities have been given quite recently by Esseen in [6] and [7] but those which interest us here are in a way more primitive than his. We shall suppose that our random variables have moments up to the third order, and we shall look for upper bounds in terms of these moments and, in § 4, of simple properties of the densities when these exist.Inequalities involving the first three moments were among the first to be found for the concentration functions of sums of independent random variables. In [9] Theorem 1, Littlewood and Offord gave an upper bound for the concentration function of SjLi e r a r where the a r are given real or complex numbers and the e r are independent random variables taking the values + 1 and -1 each with probability \. Their inequality was sharpened by Erdos [4] and extended to sums of more general random variables by Offord [11]. Offord's theorem ([11] Theorem 1), with the notation altered, may be stated as follows. THEOREM A. Let X X ,X 2 , ...,X n be one-dimensional independent random variables with E(X r ) = 0, Math. Soc. (3) 23 (1971) 489-514 Proe. London

Almost sure behavior of the critical points of random polynomials

“…The result we obtain, namely, Proposition 2.1 below, is a multidimensional version of Kolmogorov–Rogozin inequality, see, for example, Theorem 2.22 on p. 76 in [15]. This kind of estimate is not new, some variants appear, for example, in the seminal papers [4, 5] or [3] under various conditions, see also [13, 17] and the references therein. Since we were not able to find a precise reference for the exact unconditional statement that we need in both the real and complex settings, we give here a detailed proof.…”

A note on concentration functions

“…I recently worked through a proof along the lines indicated by von Bahr in order to find possible values for C t and C 2 . The proof is very similar to the one used in [2], little more than the tedious details being different, and so it will not be repeated here: I will merely remark that I found…”

“…We shall study the concentration function using characteristic functions. The possibility of doing this was discovered by Rosen ([8]), and inequalities similar to his also appear in [2] and [4]; but here we shall use one given by Petrov in [7]. If / is the characteristic function of the random variable X, then |/(0l dt.…”

“…This was originally stated for k = 3 by Esseen ([3, pp. 94, 100]), and he actually proved it for a multi-dimensional X with a unit covariance matrix; the case of a general covariance matrix was discussed by the author in [2,Lemma 8] where possible values were assigned to the quantities corresponding to C x and C 2 . The inequality (4) was stated for 2 < k < 3 by von Bahr ([1, Lemma 2]) who said that it 'can be proved in a similar way' to that used by Esseen for k = 3.…”