The concept of sub-independence is based on the convolution of the distributions of the random variables. It is much weaker than that of independence, but is shown to be sufficient to yield the conclusions of important theorems and results in probability and statistics. It also provides a measure of dissociation between two random variables which is much stronger than uncorrelatedness. Inspired by the excellent work of Jin and Lee (2014), we present certain characterizations of gamma distribution based on the concept of sub-independence

Hamedani, Gholamhossein

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Marquette Universitye-Publications@MarquetteMathematics, Statistics and Computer ScienceFaculty Research and PublicationsMathematics, Statistics and Computer Science,Department of1-1-2015Characterizations of Gamma Distribution via Sub-Independent Random VariablesGholamhossein HamedaniMarquette University, gholamhoss.hamedani@marquette.eduPublished version. Journal of the Chungcheong Mathematical Society, Vol. 28, No. 2 (2015): 187-194.DOI. © 2015 Chungcheong Mathematical Society.JOURNAL OF THECHUNGCHEONG MATHEMATICAL SOCIETYVolume 28, No. 2, May 2015http://dx.doi.org/10.14403/jcms.2015.28.2.187CHARACTERIZATIONS OF GAMMA DISTRIBUTIONVIA SUB-INDEPENDENT RANDOM VARIABLESG. G. Hamedani*Abstract. The concept of sub-independence is based on the con-volution of the distributions of the random variables. It is muchweaker than that of independence, but is shown to be sufficient toyield the conclusions of important theorems and results in prob-ability and statistics. It also provides a measure of dissociationbetween two random variables which is much stronger than uncor-relatedness. Inspired by the excellent work of Jin and Lee (2014),we present certain characterizations of gamma distribution basedon the concept of sub-independence.1. IntroductionThe concept of sub-independence is stated as follows: The rv ′s (ran-dom variables) X and Y with cdf ′s (cumulative distribution function)FX and FY are s.i. (sub-independent) if the cdf of X + Y is given by(1.1) FX + Y (z) = (FX ∗ FY ) (z) =∫RFX (z − y) dFY (y) , z ∈ R,or equivalently if and only if(1.2) ϕX + Y (t) = ϕX ,Y (t, t) = ϕX (t)ϕY (t) , for all t ∈ R,where ϕX , ϕY , ϕX+Y and ϕX,Y are cf′s (characteristic functions)of X , Y , X + Y and (X,Y ), respectively.The drawback of the concept of sub-independence in comparison withthat of independence had been that the former does not have an equiva-lent definition based on the events, which some believe, to be the naturaldefinition of independence. We have found such a definition now (seeHamedani (2013)) which is stated below for the sake of completeness.Received August 13, 2014; Accepted April 27, 2015.2010 Mathematics Subject Classification: Primary 60E05, 62E10.Key words and phrases: sub-independence, characteristic function, characteriza-tions.188 G. G. HamedaniWe shall give two separate definitions, one for the discrete case (Defini-tion 1.1) and the other for the continuous case (Definition 1.2).Let (X,Y ) : Ω → R2 be a discrete random vector with range< (X,Y ) = {(xi, yj) : i , j = 1, 2, ...} (finitely or infinitely countable).Consider the eventsAi = {ω ∈ Ω : X (ω) = xi} , Bj = {ω ∈ Ω : Y (ω) = yj}andAz = {ω ∈ Ω : X (ω) + Y (ω) = z} , z ∈ < (X + Y ) .Definition 1.1. The discrete rv ′s X and Y are s.i. if for everyz ∈ < (X + Y )(1.3) P (Az) =∑i , j ,∑xi + yj = zP (Ai)P (Bj) .To see that (1.3) is equivalent to (1.2), suppose X and Y are s.i. via(1.2), then∑i∑jeit(xi+yj) f (xi, yj) =∑i∑jeit(xi+yj) fX (xi) fY (yj) ,where f , fX and fY are probability functions of (X,Y ), X and Y re-spectively. Let z ∈ < (X + Y ), theneitz∑i , j ,∑xi + yj = zf (xi, yj) = eitz∑i , j ,∑xi + yj = zfX (xi) fY (yj) ,which implies (1.3) .For the continuous case, we observe that the half-plane H = {(x, y) :x+ y < 0} can be written as a countable disjoint union of rectangles:H = ∪∞i=1Ei × Fi,where Ei and Fi are intervals. Now, let (X,Y ) : Ω→ R2 be a contin-uous random vector and for c ∈ R , letAc = {ω ∈ Ω : X (ω) + Y (ω) < c}andA(c)i ={ω ∈ Ω : X (ω)− c2∈ Ei}, B(c)i ={ω ∈ Ω : Y (ω)− c2∈ Fi}.Definition 1.2. The continuous rv ′s X and Y are s.i. if forevery c ∈ R(1.4) P (Ac) =∞∑i=1P(A(c)i)P(B(c)i).Characterizations of Gamma Distribution 189To see that (1.4) is equivalent to (1.1), observe that (LHS of (1.4))(1.5) P (Ac) = P (X + Y < c) = P ((X,Y ) ∈ Hc) ,where Hc = {(x, y) : x+ y < c} . Now, if X and Y are s.i. thenP (Ac) = (PX × PY ) (Hc)where PX , PY are probability measures on R defined byPX (B) = P (X ∈ B) and PY (B) = P (Y ∈ B) ,and PX × PY is the product measure.We also observe that (RHS of (1.4))∞∑i=1P(A(c)i)P(B(c)i)=∞∑i=1P(X − c2∈ Ei)P(Y − c2∈ Fi)=∞∑i=1P(X ∈ Ei + c2)P(Y ∈ Fi + c2)=∞∑i=1PX × PY(Ei +c2)×(Fi +c2).(1.6)Now, (1.5) and (1.6) will be equal if Hc = ∪∞i=1{(Ei +c2)× (Fi + c2)}, which is true since the points in Hc are obtained by shifting each pointin H over to the right by c2 units and then up byc2 units.Remark 1.3. (i) Note that H can be written as a union of squaresand triangles. The triangles are congruent to 0 ≤ y < x , 0 ≤ x < 1which in turn can be written as a disjoint union of squares. For example,take [0, 1/2)× [0, 1/2) then [1/2, 3/4)× [0, 1/4) and so on. (ii) Thediscrete rv ′s X , Y and Z are s.i. if (1.3) holds for any pair and(1.7) P (As) =∑i , j , k ,∑xi + yj + zk= s∑P (Ai)P (Bj)P (Ck) .For p variate case we need 2p−p−1 equations of the above form. (iii)The representation (1.2) can be extended to the multivariate case aswell. (iv) For a detailed treatment of the concept of sub-independence,we refer the interested reader to Hamedani (2013) .2. CharacterizationsCharacterizations of distributions are important to many researchersin the applied fields. An investigator will be vitally interested to knowif their model fits the requirements of a particular distribution. To this190 G. G. Hamedaniend, one will depend on the characterizations of this distribution whichprovide conditions under which the underlying distribution is indeedthat particular distribution. Various characterizations of distributionshave been established in many different directions. In this work, severalcharacterizations of gamma distribution are presented. Inspired by theexcellent work of Jin and Lee (2014), we would like to establish thefollowing characterizations of gamma distribution based on the conceptof sub-independence.Proposition 2.1. Let X and Y be positive s.i.i.d. (sub-independentand identically distributed) rv ′s with an absolutely continuous cdf Fand E[X2]<∞. The rv X has a gamma distribution if and only ifthere exists a γ ∈ (0, 1/4) such that for t ∈ R,(2.1)(ϕ′ (t))2ϕ′′1 (t)= γ ,where ϕ (t) and ϕ1 (t) are cf′s of X and X + Y respectively.Proof. If X has a gamma distribution, then clearly (2.1) holds.Now, assume (2.1) holds. Since X and Y are s.i. , from (1.2) wehave(2.2) ϕ1 (t) = (ϕ (t))2 , t ∈ R.Differentiating both sides of (2.2) twice, we arrive at(2.3) ϕ′′1 (t) = 2{ϕ (t)ϕ′′ (t) +(ϕ′ (t))2}.Now, from (2.1) and (2.3) we obtainϕ (t)ϕ′′ (t) =(1− 2γ2γ)((ϕ′ (t))2),or(2.4)ϕ′′ (t)ϕ′ (t)=(1− 2γ2γ)ϕ′ (t)ϕ (t),or(2.5) ϕ′ (t) = iE (X) (ϕ (t))1−2γ2γ ,in which the initial conditions ϕ (0) = 1 and ϕ′ (0) = iE (X) are used.Solving (2.5) for ϕ (t) , we have, in view of 0 < γ < 1/4ϕ (t) =(1− i(1− 4γ2γ)E (X) t)−( 2γ1−4γ), t ∈ R ,Characterizations of Gamma Distribution 191which is the cf of a gamma distribution with parameters(2γ1−4γ)and(1−4γ2γ)E (X) .Under the assumption of independence of the rv ′s X and Y , wehave the following proposition.Proposition 2.2. Let X and Y be positive i.i.d. rv ′s with anabsolutely continuous cdf F and E[X2]< ∞. The rv X has agamma distribution if and only if there exists a γ ∈ (0, 1/4) such thatfor t ∈ R,(2.6)(E [X exp {itX}])2E[(X + Y )2 exp {it (X + Y )}] = γ.Proof. If X has a gamma distribution, then clearly (2.6) holds.Now, assume (2.6) holds and observe that∫ ∞0∫ ∞0(x− y)2 exp {it (X + Y )} dF (x) dF (y)=∫ ∞0∫ ∞0(x+ y)2 exp {it (X + Y )} dF (x) dF (y)− 4(∫ ∞0x exp {itX} dF (x))2= (1− 4γ)∫ ∞0∫ ∞0(x+ y)2 exp {it (X + Y )} dF (x) dF (y) .(2.7)Now, expressing (2.7) in terms of cf ′s , we have−2ϕ (t)ϕ′′ (t) + 2 (ϕ′ (t))2 = −2 (1− 4γ){ϕ (t)ϕ′′ (t) + (ϕ′ (t))2} ,from which we arrive at the basic differential equation (2.4).Proposition 2.3. Let X and Y be positive s.i.i.d. rv ′s withan absolutely continuous cdf F and E[X2]< ∞ and let (X + Y )and(X−YX+Y)2be s.i.i.d.. The rv X has a gamma distribution if andonly if there exists a γ ∈ (0, 1/4) such that for t ∈ R,(2.8)(ϕ′1 (t))2ϕ′′2 (t)= γ ,where ϕ1 (t) and ϕ2 (t) are cf′s of (X + Y ) and (X + Y )+(X−YX+Y)2,respectively.192 G. G. HamedaniProof. Observe that in view of sub-independence of (X + Y ) and(X−YX+Y)2,we have(2.9) ϕ2 (t) = (ϕ1 (t))2 .It is easy to see from (2.9) that if X has a gamma distribution, (2.8)holds. Now, assume that (2.8) holds and note that differentiating(2.9) twice and using (2.8), we have(1− 2γ) (ϕ′1 (t))2 = 2γϕ1 (t)ϕ′′1 (t) ,orϕ′′1 (t)ϕ′1 (t)=(1− 2γ2γ)ϕ′1 (t)ϕ1 (t)from which we arrive atϕ1 (t) =(1− i(1− 4γ2γ)E (X + Y ) t)−( 2γ1−4γ), t ∈ R,i.e., (X + Y ) has a gamma distribution with parameters(2γ1−4γ)and(1−4γ2γ)E (X + Y ) . Since X and Y are s.i.i.d. , their common cfisϕ (t) =(1− i(1− 4γ2γ)E (X + Y ) t)−( γ1−4γ), t ∈ R,i.e., X has a gamma distribution with parameters(γ1−4γ)and(1−4γ2γ)E (X + Y ) .Under the assumption of independence of the rv ′s X and Y wehave the following proposition.Proposition 2.4. Let X and Y be positive i.i.d. rv ′s with anabsolutely continuous cdf F and E[X2]<∞ and let (X + Y ) and(X−YX+Y)2be s.i.i.d. The rv X has a gamma distribution if and onlyif there exists a γ ∈ (0, 1) such that for t ∈ R,(2.10)E[{(X + Y )−(X−YX+Y)2}2exp{it[(X + Y ) +(X−YX+Y)2]}]E[{(X + Y ) +(X−YX+Y)2}2exp{it[(X + Y ) +(X−YX+Y)2]}] = γ.Characterizations of Gamma Distribution 193Proof. If X has a gamma distribution, then clearly (2.10) holds.Now, if (2.10) holds, then we have∫ ∞0∫ ∞0{(X + Y )−(X − YX + Y)2}2× exp{it[(X + Y ) +(X − YX + Y)2]}dF (x) dF (y)=∫ ∞0∫ ∞0{(X + Y ) +(X − YX + Y)2}2× exp{it[(X + Y ) +(X − YX + Y)2]}dF (x) dF (y)− 4(∫ ∞0∫ ∞0(X + Y ) exp {it (X + Y )} dF (x) dF (y))2= γ∫ ∞0∫ ∞0{(X + Y ) +(X − YX + Y)2}2× exp{it[(X + Y ) +(X − YX + Y)2]}dF (x) dF (y) ,or(1− γ)∫ ∞0∫ ∞0{(X + Y ) +(X − YX + Y)2}2× exp{it[(X + Y ) +(X − YX + Y)2]}dF (x) dF (y)(2.11)= 4(∫ ∞0∫ ∞0(X + Y ) exp {it (X + Y )} dF (x) dF (y))2.Now, expressing (2.11) in terms of cf ′s , we have(1− γ)ϕ1 (t)ϕ′′1 (t) + (1− γ)(ϕ′1 (t))2= 2(ϕ′1 (t))2,orϕ′′1 (t)ϕ′1 (t)=(1 + γ1− γ)ϕ′1 (t)ϕ1 (t),194 G. G. Hamedanifrom which, we obtainϕ1 (t) =(1− i(2γ1− γ)E (X + Y ) t)−( 1−γ2γ), t ∈ R,i.e., (X + Y ) has a gamma distribution with parameters(1−γ2γ)and(2γ1−γ)E (X + Y ) . Since X and Y are s.i.i.d. , their common cf isϕ (t) =(1− i(2γ1− γ)E (X + Y ) t)−( 1−γ4γ), t ∈ R,i.e., X has a gamma distribution with parameters(1−γ4γ)and(2γ1−γ)E (X + Y ) .Remark 2.5. Similar Propositions can be stated for Sn =∑ni=1Xiand∑mi=1X2iS 2n, 1 ≤ m < n , where Xi ′s are (s.i.i.d. or i.i.d.) and Snand∑mi=1X2iS 2n, m = 1, 2, ..., n are s.i.i.d.References[1] G. G. Hamedani, Sub-independence: An Expository Perspective, Commun.Statist. Theory-Methods 41 (2013), 3615-3638.[2] H.-W. Jin and M.-Y. Lee, Characterizations of the Gamma Distribution byIndependence Property of Random Variables, J. of Chungcheong Math. Soc.27 (2014), no. 2, 157-163.*Department of MathematicsMarquette UniversityMilwaukee, WI 53201-1881E-mail : gholamhoss.hamedani@marquette.edu

Characterizations of Gamma Distribution via Sub-Independent Random Variables

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Characterizations of Gamma Distribution via Sub-Independent Random Variables

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