A Note on the Sum of Correlated Gamma Random Variables

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1 1 A Note on the Sum of Correlated Gamma Random Variable Joé F Pari Abtract arxiv: v1 [cit] 2 Mar 2011 The um of correlated gamma random variable appear in the analyi of many wirele communication ytem, eg in ytem under Nakagami-m fading In thi Letter we obtain exact expreion for the probability denity function PDF and the cumulative ditribution function CDF of the um of arbitrarily correlated gamma variable in term of certain Lauricella function Index Term Gamma variate, Nakagami-m fading, Outage Probability, Lauricella Function I INTRODUCTION Many of the performance analyi problem in the cope of wirele communication theory require determination of the tatitic of the um of the quared envelope of Nakagami-m faded ignal or, equivalently, the um of gamma random variable ince the quare of a Nakagami-m random variable follow a gamma ditribution [1] Some expreion are available in literature for the probability denity function PDF of the um of gamma random variable, eg ee [1] and the reference cited herein Thee expreion are frequently in the form of infinite erie, including the general expreion for arbitrary correlation provided in [1] In thi Letter we reviit the reult derived in [1]; we how that, under the ame aumption, the PDF and the cumulative ditribution function CDF of the um of correlated gamma random variable can be expreed in a compact form by certain Lauricella function To the bet of the author knowledge, the expreion obtained here are novel *Thi work i partially upported by the Spanih Government under project TEC /TCM and by AT4 wirele

2 2 II ANALYTICAL RESULTS For clarity we will ue the notation adopted in [1] We ay X follow a gamma ditribution with parameter > 0 and β > 0 if the PDF of X i given by p X x = x 1 e x/β Ux, 1 β Γ where Γ i the gamma function and U i the unit tep function The horthand notation X G,β will be ued to denote that X i gamma ditributed with parameter and β The key idea of thi Letter i the following In [1] the author extended the Mochopoulo Theorem [2] Interetingly, they oberved the imilarity between the MGF of the correlated cae and the independent cae Then, the Mochopoulo technique of inverting the MGF wa again adopted for the correlated cae Here, we tart from the MGF of the correlated cae, but intead of uing the Mochopoulo technique, we etablih a connection between the MGF and certain Lauricella function The firt who etablihed a connection of thi type wa Kabe [3], within the context of independent gamma random variable The mathematically precie tatement are given below Lemma 1: Let {X n } N be a et of N correlated not neceary identically ditributed gamma random variable with parameter and β n, repectively, [ie, X n G,β n ] and let ρ ij denote the correlation coefficient between X i and X j, ie, ρ ij = ρ ji = covx i,x j varxi varx j, 0 ρ ij 1 i,j = 1,2,,N 2 then the CDF of Y = N X n can be expreed a y N F Y y = deta Γ1+N Φ N 2,,;1+N; yλ1,, yλn, 3 where Φ N 2 i the confluent Lauricella function [4][5], and { } N matrix A = DC where D i the N N diagonal matrix with the entrie {β n } N are the eigenvalue of the and C i the

3 3 N N poitive definite matrix defined by 1 ρ12 ρ21 1 C = ρn1 1 ρ1n ρ2n 4 The PDF of Y i given by { } y 1+N f Y y = deta ΓN Φ N 2,,;N; yλ1,, yλn 5 Proof: See Appendix I The expreion derived in Lemma 1 are compact and can be frequently reduced to impler form uing the propertie of the function Φ N 2 In particular, ince Φ F 1, [ie, equivalent to the confluent hypergeometric function] one can check that for = m, β 1 = γ/m and N = 1 the expreion derived here reduce to the well-known CDF and PDF of the quare of a Nakagami-m random variable For reduction formula, integral repreentation and integral involving Φ N 2, the reader hould refer to [4] Note that the CDF expreion given Lemma 1 allow u to compute the outage probability of maximal ratio combining MRC over correlated Nakagami-m fading channel III CONCLUSIONS In thi Letter, compact expreion have been derived for the um of arbitrarily correlated gamma random variable Such expreion have both theoretical and practical value, and are applicable in a vat range of wirele communication problem APPENDIX A PROOF OF LEMMA 1 For an arbitrary function φx we denote the Laplace tranform a L[φx;] A in [1], we define the MGF of Y a M Y = E[e y ] = L[f Y y; ] We know that the MGF of Y i given

4 4 by [1] M Y = 1 6 Therefore, we can write the CDF of Y in the following form L[F Y y;] = 1 L[f Y y;] = Γ 1+ N i=1 = Γ 1+ N N Then, after identifying 7 with [5, p 222, eq 5], the CDF of Y i obtained To derive the expreion for the PDF we can write L[f Y y;] = = N Γ N Γ i=1 N Again, after identifying thi expreion with [5, p 222, eq 5], the PDF of Y i obtained 7 8 REFERENCES [1] M-S Alouini, A Abdi and, M Kaveh, Sum of gamma variate and performance of wirele ommunication ytem over Nakagami fading channel, IEEE Tran Veh Technol, vol 50, pp , Nov 2001 [2] P G Mochopoulo, The ditribution of the um of independent gamma random variable, Ann Int Statit Math Part A, vol 37, pp , 1985 [3] D G Kabe, On the exact ditribution of a cla of multivariate tet criteria, Ann Math Statit, vol 33, pp , 1962

5 5 [4] H Exton, Multiple Hypergeometric Function and Application, Halted Pre John Wiley & Son, 1976 [5] A Erdelyi, Table of Integral Tranform, vol I, McGraw Hill, New York, 1954

On the Multivariate Nakagami-m Distribution With Exponential Correlation

1240 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 8, AUGUST 2003 On the Multivariate Nakagami-m Distribution With Exponential Correlation George K. Karagiannidis, Member, IEEE, Dimitris A. Zogas,