Characterizations of Levy Distribution via Sub- Independence of the Random Variables and Truncated Moments

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1 Marquette University MSCS Faculty Research Publications Mathematics, Statistics Computer Science, Department of 7--5 Characterizations of Levy Distribution via Sub- Independence of the Rom Variables Truncated Moments Gholamhossein G. Hamedani Marquette University, M. Ahsanullah Rider University-Lawrenceville Seyed Morteza Najibi Persian Gulf University Published version. Pakistan Journal of Statistics, Vol. 3, No. 4 ( July 5): Publisher Link. Pakistan Journal of Statistics. Used with permission.

2 Pak. J. Statist. 5 Vol. 3(4), CHARACTERIZATIONS OF LEVY DISTRIBUTION VIA SUB-INDEPENDENCE OF THE RANDOM VARIABLES AND TRUNCATED MOMENTS G.G. Hamedani, M. Ahsanullah S.M. Najibi 3 Department of Mathematics, Statistics Computer Science Marquette University, Milwaukee, WI , USA gholamhoss.hamedani@marquette.edu Department of Management Sciences, Rider University Lawrenceville, NJ. 8648, USA. ahsan@rider.edu 3 Department of Statistics, Persian Gulf University Bushehr, Iran, sm_najibi@sbu.ac.ir ABSTRACT The concept of sub-independence is based on the convolution of the distributions of the rom variables. It is much weaker than that of independence, but is shown to be sufficient to yield the conclusions of important theorems results in probability statistics. It also provides a measure of dissociation between two rom variables which is much stronger than uncorrelatedness. Following Ahsanullah Nevzorov (4), we present certain characterizations of Levy distribution based on: (i) the sub-independence of the rom variables; (ii) a simple relationship between two truncated moments; (iii) conditional epectation of certain function of the rom variable. In case of independence, characterization (i) reduces to that of Ahsanullah Nevzorov (4). KEYWORD Sub-independence, Characterization, Levy distribution, Truncated moments.. INTRODUCTION In designing a stochastic model for a particular modeling problem, an investigator will be vitally interested to know if their model fits the requirements of a specific underlying probability distribution. Consequently, the investigator relies on the characterizations of the selected distribution. Generally speaking, the problem of characterizing a distribution is an important problem which has recently attracted the attention of many researchers, resulting in various characterizations reported in many different directions. The present work deals with the characterizations of Levy distribution. These characterizations are based on: (i) the sub-independence of rom variables; (ii) a simple relationship between two truncated moments; (iii) conditional epectation of certain function of the rom variable. For the sake of completeness we will state below a few definitions related to the concept of sub-independence. The concept of sub-independence is stated as follows: The 5 Pakistan Journal of Statistics 47

3 48 Characterizations of Levy Distribution via Sub-Independence rv ' s (rom variables) Y with cdf ' s (cumulative distribution functions) F Y are si.. (sub-independent) if the cdf of Y is given by F Y z = F FY z = F z y dfy y, z, (.) or equivalently if only if t = t, t = t t, for all t, (.) Y, Y Y where, Y, Y Y, are cf ' s (characteristic functions) of, Y, Y Y,, respectively. The equations (.) (.) above are in terms of cdf cf. The definition of sub-independence in terms of events, similar to that of independence, is as follows. Let, : Y be a discrete rom vector with range, Y = ( i, y j): i, j =,,... (finitely or infinitely countable). Consider the events Ai z, = : = i B = : Y = y A = : Y = z, z Y, where, as usual, sample space. Definition. The discrete j is the value of the rom variable at the point in the rv ' s Y are.. z P Ai PB j P A i, j, i y j = z si if for every z Y =. (.3) For the continuous case, we observe that the half-plane can be written as a countable disjoint union of rectangles: i= i i H = E F, where Ei i vector for c F are intervals. Now, let, :, let A = : Y < c c j F H =, y : y < Y be a continuous rom c c c c Ai = : Ei, Bi = : Y Fi.

4 Hamedani, Ahsanullah Najibi 49 Definition. The continuous rv ' s Y are si.. if for every c c c P Ac = PAi P Bi. (.4) i= Remarks.3 a The discrete rv ' s, Y Z are si.. if (.3) holds for any pair s i j k P A P A P B P C. (.5) i, j, k, y z s i j k p For p variate case we need p equations of the above form. (b) The representation (.) can be etended to the multivariate case as well. (c) For a detailed treatment of the concept of sub-independence, we refer the interested reader to Hamedani (3). Ahsanullah Nevzorov (4) consider the Levy distribution with parameters Lev(, ) whose pdf (probability density function) cf are respectively given by 3/ f = f ;, = e, >, i t t = e. They presented the following characterization of Lev, distribution. (.6) Theorem.4,, be independent identically distributed absolutely continuous Let 3 rom variables with cdf for all >. Then has the Lev, distribution. F pdf f. We assume F = F > are identically distributed if only if F 3 4 In subsection. below, we present similar characterization based on the concept of sub-independence which in turn is stronger than Theorem.4.. CHARACTERIZATION RESULTS In this section we present characterizations of Lev, directions (i)-(iii) mentioned in the Introduction. distribution in three different. Characterizations via Sub-Independence of the Rom Variables Lev, distribution. We present here two Lemmas characterizing

5 4 Characterizations of Levy Distribution via Sub-Independence Lemma..,, be identically distributed continuous rom variables with cdf Let 3 F pdf f. We assume F = > d ) ~ Lev, e ) the rom variables 4 3 / 4 ~ Lev,. Proof: In view of e ), we have i.e., F for all >. If 3 4 it / 4 it t t, 3 /4 /4 t e e Lev 3 /4 ~,. are si.., then Lemma...,, be identically distributed continuous rom variables with cdf Let 3 F pdf f. We assume F = > rom variables 3 /4 n =,,..., the rom variables 3 are.. F for all >. If f ) the are identically distributed i ) for each n n Proof: In view of f ) i ) (for n =), we have t t = /, in view of i ), we arrive at n n si, then Lev t = t / =... = t /, n =,,... ~,. Now, the rest follows as in the proof of Theorem.4 (Ahsanullah Nevzotov (4), page ).. Characterizations based on Two Truncated Moments In this subsection we present characterizations of Levy distribution in terms of a simple relationship between two truncated moments. We like to mention here the works of Glänzel (987,99), Glänzel Hamedani () Hamedani (,6) in this direction. Our characterization results presented here will employ an interesting result due to Glänzel (987) (Theorem.. below). The advantage of the characterizations

6 Hamedani, Ahsanullah Najibi 4 given here is that, cdf F need not have a closed form are given in terms of an integral whose integr depends on the solution of a first order differential equation, which can serve as a bridge between probability differential equation. Theorem.. Let,,P be a given probability space let I =, a< b a=, b= might as well be allowed. Let : I a b be an interval for some be a continuous rom variable with the distribution function F let g h be two real functions defined on I such that E g = E h, I, is defined with some real function. Assume that g,, C I h C I F is twice continuously differentiable strictly monotone function on the set I. Finally, assume that the equation h = g has no real solution in the interior of I. Then F is uniquely determined by the functions g, h, particularly ' u uh u F = C ep s u du, a g u where the function s is a solution of the differential equation constant, chosen to make = I df. ' h s '= hg C is a We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence, in particular, let us assume that there is a sequence n of rom variables with distribution functions F n such that the functions g n, h n n n satisfy the conditions of Theorem.. let gn g, hn h for some continuously differentiable real functions g h. Let, g finally, be a rom variable with distribution F. Under the condition that n hn are uniformly integrable the family F n is relatively compact, the sequence n converges to in distribution if only if E = g. E h n converges to, where This stability theorem makes sure that the convergence of distribution functions is reflected by corresponding convergence of the functions g, h, respectively. It guarantees, for instance, the convergence of characterization of the Wald distribution to that of the Lévy-Smirnov distribution if, as was pointed out in Glänzel Hamedani ().

7 4 Characterizations of Levy Distribution via Sub-Independence Remarks.. In Theorem.., the interval I need not be closed since the condition is only on the interior of I. Proposition..3 Let :, be a continuous rom variable let h = = / g e / for,. The pdf of is (.6) if only if the function defined in Theorem.. has the form = e, >. Proof: Let have density (.6), then finally F E h = e, >, F E g = e, >, / h g = e > for >. Conversely, if is given as above, then ' h e s ' = =, >, h g e hence s = log e, >. Now, in view of Theorem.., has density (.6).

8 Hamedani, Ahsanullah Najibi 43 Corollary..4 Let :, be a continuous rom variable let h be as in Proposition..3. The pdf of is.6 if only if there eist functions g defined in Theorem.. satisfying the differential equation ' h e =, >. h g e Remarks..5 A) The general solution of the differential equation in Corollary..4 is 3/ = e e g d D, for >, where D is a constant. One set of appropriate functions is given in Proposition..3 with D =. B) Clearly there are other triplets of functions hg,, satisfying the conditions of Theorem....3 Characterizations based on Certain Function of the Rom Variable We assume, without loss of generality, that the parameter of Levy distribution is. We start with the following simple lemma. Lemma.3. :, be a continuous positive rom variable with twice Let differentiable strictly increasing cdf F corresponding pdf f >. Let h be a function on, with E h f < h h h h d differentiable function on, such that eists. Then f E h = h, > F implies = ep h u h u f c du, h u > where c is chosen so the f d =. with be a

9 44 Characterizations of Levy Distribution via Sub-Independence Proof: The proof is straightforward will be omitted. Proposition.3. :, be a positive continuous rom variable with twice Let differentiable strictly increasing cdf f >. Then F corresponding pdf f with k f E = h, >, F (.3.) where >. k k! 3 k j = = k j if only if k j! h 3/ f = e, Proof: If has pdf 3/ f = e, >, then k k 3/ F E = u u e u du k = u e udu k = k t t e dt k k k j! = k e j= k j! 3/ = h e, 3 k where k k! k j h = j=. Thus, (.3.) holds. k j! Now, suppose (.3.) holds with k 3 k k! k j j= h =. k j!

10 Hamedani, Ahsanullah Najibi 45 Then k k k! 3 k j h = k j ( ) j= k j! = k j ( ) ( ) j= k j! j= k j! k k k! k j 3 k k! k j k 3 = h h, after a couple of simplification steps. Thus, we have k 3 h = h. Now, using Lemma.3. the fact that 3/ f = e, >. f d =, we obtain ACKNOWLEDGEMENTS The authors are grateful to the referees for their valuable suggestions which greatly improved the presentation of the content of the paper. REFERENCES. Ahsanullah, M. Nevzorov, V.B. (4). Some inferences on the Levy distribution. JSTA, 3, Glänzel, W. (987). A characterization theorem based on truncated moments its application to some distribution families, Mathematical Statistics Probability Theory (Bad Tatzmannsdorf, 986), Vol. B, Reidel, Dordrecht, Glänzel, W. (99). Some consequences of a characterization theorem based on truncated moments. Statistics,, Glänzel, W. Hamedani, G.G. (). Characterizations of univariate continuous distributions. Studia Sci. Math. Hungar., 37, Hamedani, G.G. (). Characterizations of univariate continuous distributions-ii. Studia Sci. Math. Hungar., 39, Hamedani, G.G. (6). Characterizations of univariate continuous distributions-iii, Studia Sci. Math. Hungar., 43, Hamedani, G.G. (3). Concept of sun-independence: an epository perspective. Commun. Statist. Theory-Methods,