Central limit theorem for signed distributions

Abstract

Abstract. This paper contains an improved version of existing generalized central limit theorems for convergence of normalized sums of independent random variables distributed by a signed measure. It is shown that under reasonable conditions, the normalized sums converge in distribution to "higher-order" analogues of the standard normal random variable, in the sense that the density of the limiting signed distribution is the fundamental solution of a higher-order parabolic partial differential equation that is a generalization of the heat equation.1. Introduction. Probability theory is often thought of as the study of nonnegative bounded measures. However, several theorems usually considered to be purely probabilistic are, in fact, true in more general contexts. Specifically, several authors have extended such results to signed measures. For example, Hochberg [4] derives generalizations of the Brownian motion or Wiener process to higher-order signed stochastic processes, an analogue of the Itô stochastic calculus for these signed processes, and a probabilistic-style analysis of the spectral properties of higherorder elliptic operators.Similarly, the central limit theorem for convergence of normalized sums of random variables to the Gaussian distribution has been extended to the case where the random functions are distributed by a signed measure. Zukov [8] derives such a theorem for difference operators in a study of a problem in numerical analysis. Studnev [6] states, without proof, other generalizations, with an application to probability theory itself. Hersh [3] gives a more complete study of generalized central limit theorems, though his results do not apply to the cases n = 0 (mod 2) in what follows; that is, Hersh's theorems exclude the cases where the signed distribution has a density function which is the fundamental solution of the parabolic partial differential equation (2) below whose order is a multiple of four. Our techniques include these cases.In an earlier paper (Hochberg [4]), we derived two such central limit theorems, which we restate in the following two propositions. Here

An Itô calculus for a class of limit processes arising from random walks on the complex plane

“…By using formula (15), the remaining terms in the sum (corrisponding to the h−ple (m N , m 2N , ..., m hN ) with m N < h) are bounded by…”

“…It is important to recall that due to the particular conditions necessary for the generalization of the Kolmogorov existence theorem for the limit of a projective system of complex measures (see [30]), in the case of Krylov-Hochberg process, a well defined signed measure on R [0,t] cannot exist and the "integrals" realizing the Feynman-Kac formula for equation (3) are just formal expressions which cannot make sense in the framework of Lebesgue integration theory but are to be meant as limit of a particular approximating sequence. However, even taking into account these technical problems, an analog of the arc-sine law [14,16,19], of the central limit theorem [15,29] and of Itô formula and Ito stochastic calculus [14,25] have been developed for the (finite additive) Krylov-Hochberg signed measure. For a extensive discussion of these problems in the framework of a generalized integration theory on infinite dimensional spaces as well as for a unified view of probabilistic and complex integration see [1,2].…”

Probabilistic Representations for the Solution of Higher Order Differential Equations

“…giving the semigroup −( /8)Δ 2 in terms of a Gaussian expectation, with respect to the time variable ∈ R, of the Schrödinger group (Δ/2) . By applying (12) to an initial datum 0 ∈ (R) one can write the following chain of equalities:…”

“…It is worthwhile to mention that an analogous of the arcsine law [10,11], of the central limit theorem [12], and of Ito formula and Ito stochastic calculus [10,13] have been proved for the (finite additive) signed measure 2 . Moreover, a Feynman-Kac formula has been proved [9][10][11], for the representation of the solution of the Cauchy problem (3) in the case where is a bounded piecewise continuous function and for an initial datum 0 ∈ 2 , by realizing formula (4) as limit of finite dimensional cylindrical approximations [14].…”

High order heat-type equations and random walks on the complex plane

“…For this reason the integral in is not defined in Lebesgue sense, but is meant as limit of finite dimensional cylindrical approximations [3]. It is worthwhile to mention that an analogous of the arc-sine law [14,16], of the central limit theorem [15] and of Itô formula and Itô stochastic calculus [14] have been proved for the Krylov-Hochberg pseudo-process. In the same line, we shall mention the papers by Nishioka [28,29].…”