Abstract
Characterization problems in probability are studied here. Using the characteristic function of an additive convolution we generalize some known characterizations of the normal distribution to stable distributions. More precisely, if a distribution of a linear form depends only on the sum of powers of the certain parameters, then we obtain symmetric stable distributions.
1. Introduction
The original motivation for this paper comes from a desire to understand the results about characterization of normal distribution which were shown in [1]. In this paper, the author provides characterizations of the normal distribution using a certain invariance of the noncentral chi-square distribution. More precisely, let statistic have a distribution which depends only on , with , , where and are independent random vectors with all moments, and are nondegenerate; then are independent and have the same normal distribution with zero means and for . The proof of the above theorem is divided into two parts: first, it is proved that this result holds for two random variables. Second, it is shown using the properties of multidimensional normal distribution. The additional moment assumption is due to the fact that the author uses a method of cumulants. An alternative method of proof (more direct and straightforward one) allows us to weaken some of the technical assumptions used in the above references and generalize it to a symmetric stable distribution. The paper is organized as follows. In Section 2 we review basic facts about characteristic function. Next in Section 3 we state and prove the main results.
2. A Characteristic Function
In this paper we denote by a probability measure of random variable . If is a random variable defined on a probability space , then the expected value of , denoted by , is defined as the Lebesgue integral: A characteristic function is simply the Fourier transform, in probabilistic language. The characteristic function of a probability measure on is the function : When we speak of the characteristic function of a random variable , we have the characteristic function of its distribution in mind. Note, moreover, that Apparently, it is not accidental that the characteristic function encodes the most important information about the associated random variables. The underlying reason may well reside in the following three important properties: (i)The Gaussian distribution has the characteristic function .(ii)The symmetric -stable distribution has the characteristic function , where and . For the special cases of parameter , we get(1)the upper bound corresponding to the normal distribution,(2) corresponding to the Cauchy distribution,(3)for the distribution reduces to a Lévy distribution.(iii)Random variables are independent if and only if for all the joint characteristic function (i.e., the linear combination of the ’s) satisfies
3. The Characterization Theorem
The main result of this paper is the following characterization of the symmetric -stable distribution in terms of independent random vectors.
Theorem 1. Let and be independent random vectors, where the distribution of is not Dirac measure and let statistic have a distribution which depends only on , for all , , being an arbitrary random variable, and . Then are independent and have the same symmetric -stable distribution. Additionally, if and then for .
Proof. We write , where . We focus on characteristic function of . Then for we have Since the hypothesis implies that left side of (5) does not depend on , then does not depend on . In particular we have that the distribution of a statistic depends on only. Let . Because of the independence of and we may writeEvaluating (8) first when and then when , we get and , respectively. Substituting this into (8), we seeNote that is continuous in , which implies (see Aczel [2], page 31), and so we have . Thus we have actually proved that have the same symmetric -stable distribution. But since we know that the distributions of are symmetric stable, the independence of random variables follows from the observation that If and then and because of the independence of we may write because for and . This implies that linear combination is constant on sphere which gives .
The above consideration gives us the following results of Ejsmont [1] and Cook [3], respectively.
Corollary 2 (the main result of Ejsmont [1]). Let , and be independent random vectors with all moments, where are nondegenerate, and let statistic have a distribution which depends only on for all and . Then are independent and have the same normal distribution with zero means and for .
Corollary 3 (the main result of Cook [3]). Let and be independent random vectors, where are nondegenerate, and let statistic have a distribution which depends only on , and . Then are independent and have the same normal distribution with zero means.
Proof. If we put and in Theorem 1 then we get This means that the distribution of depends only on , which by Theorem 1 implies the statement.
Here we state the Herschel-Maxwell theorem in modern notation (see, e.g., [4] or [5]). This theorem can be also obtained from Theorem 1 by considering and as well as and (the proof is left to the reader).
Theorem 4. Let , be independent random variables and real numbers such that . Then , are normally distributed with zero means if and only if is distributed identically as for any .
Open Problem. Kagan and Letac [6] formulate the following theorem: Let be a fixed integer . Let be independent identically distributed random variables. In the Euclidean space consider the linear subspace , that is, the set . Then the following characterizations hold: If for all the distribution of the random variable depends only on , then ’s are normally distributed.
A key role in the proof of these results is played by Marcinkiewicz’ theorem: if is a polynomial and is the characteristic function of some probability distribution, then the degree of is less than or equal to two. Finally, we present the conjecture (Theorem 1 cannot be applied here).
Conjecture 5. Let and be independent random vectors, where are nondegenerate, and the distribution of the random variable depends only on , where , and then ’s are normally distributed independent random variables.
Competing Interests
The author declares having no competing interests.
Acknowledgments
The author would like to thank M. Bożejko for several discussions and helpful comments during the preparation of this paper. The work was partially supported by the Narodowe Centrum Nauki, Grant no. 2014/15/B/ST1/00064, and by the Austrian Science Fund (FWF) Project no. P 25510-N26.