### Journal Menu >> Open Journal of Discrete Mathematics, 2011, 1, 127-135 doi:10.4236/ojdm.2011.13016 Published Online October 2011 (http://www.SciRP.org/journal/ojdm) Copyright © 2011 SciRes. OJDM The Equilibrium Distribution of Counting Random Variables Shuanming Li Centre for Actuarial Studies, Department of Economics, the University of Melbourne, Australia E-mail: shli@unimelb.edu.au Received July 1, 2011; revised August 3, 2011; accepted August 15, 2011 Abstract We study the high order equilibrium distributions of a counting random variable. Properties such as moments, the probability generating function, the stop—loss transform and the mean residual lifetime, are derived. Ex-pressions are obtained for higher order equilibrium distribution functions under mixtures and convolutions of a counting distribution. Recursive formulas for higher order equilibrium distribution functions of the -family of distributions are given. ,,0ab Keywords: Counting Random Variable, Equilibrium Distribution, Stop-Loss Transform, Mean Residual Life, Family, Recursive Formulas, Probability Generating Function ,,0ab1. Introduction Recently, there has been much attention given to higher order equilibrium distributions associated with a given distribution function (d.f.), see e.g., Fagiuoli and Pellerey [1,2], Nanda, Jain and Singh , Hesselager, Wang and Willmot  and the references therein. Equilibrium dis-tributions arise naturally in ruin theory and play an im-portant in various settings. The first order equilibrium distribution of a claim size d.f., in classical risk theory, can be interpreted as the distribution of the amount of the first drop below the initial reserve, given there is such a drop (see for instance Bowers et al. , Chapter 12). Many results on the mo-ments of the time to ruin, the surplus before ruin and the deficit at ruin, heavily depend on the equilibrium distri-bution of the claim size d.f. [see Lin and Willmot [6,7] for details]. Some classifications of reliability distributions are based on properties of higher order equilibrium distribu-tions. Whence, bounds for the right tail of the total claims distribution and ruin probabilities, can be obtained from the properties of equilibrium distributions associated with the single claim size d.f., see [7-9]. Although much attention has been paid to the equilib-rium distributions associated with a given d.f., most re-sults are for continuous random variables. Instead, we discuss higher order equilibrium distributions associated with a discrete probability function (p.f.). Throughout the paper, =0,1,2, and =1,2, . 2. Notation and Definitions Let X be a non-negative r.v. taking integer values, with probability function (p.f.) survival function  ==pxPX x,1= ,yxP=>xPXx pyx  and -th moment n=.nnEXConsider the equilibrium distribution of p, defined as   11111:= =,.yxPxpxpy x Now, define  := nnEXnxx to be the n-th factorial moment of X, where de-notes the n-th factorial power of x and It is well known in summation calculus (see e.g. Hamming =1x xn0=1.x1, p.182) that  11=1=,1nnynkxyxkn for ,n ,xy , and .xy Hence the n-th factorial moment 1: n of is given by 1p 11:11112=111==1 =,1,nnnxxyynyx1xxpxx pypy xn 128 S. M. LI  112111111==11 =,for=.1nnynypy nnn1 (1) Similarly, the probability generating function (p.g.f.) of the equilibrium distribution is given by 1p  1101ˆ1ˆ==,11xxpsps spxss 1, with and its survival function is 1ˆ1=1p  11111111==1 =1.yxyx kykxPxp ypkpk kx 1 (2) Now define the equilibrium distribution of 1 or equivalently, the second order equilibrium distribution of ,p:p  121:111:1 1=1 =1,,yxPxpxyx pyx  where 1:1 is the first order moment of 1. The facto-rial moments of are obtained as in (1) to be p2p1: 11: 12:1:1 1: 1==11nnnnn. Then the p.g.f. of is given by 2px 12201:1ˆ1ˆ==,1<1xxpsps spxss<1, with and the corresponding survival function 2ˆ1=1,p221=.kxPx pk Define similarly the subsequent equilibrium distribu-tions of from the third order ,p32:12=1px Px up to the n-th order 1:1 1=1nnnpxP x for where the following theorem gives an expres-sion for ,xnPx and 1.npxTheorem 1 The survival function ,nPx of the n-th order equilibrium distribution can be expressed as np  11:1=01=1!nnnyxllPxpy yxxn,, (3)  11 =1,nyxnpy yx (4)  1111=1nnyxnnpx pyyx , (5) where :1l is the mean of -th order equilibrium dis-ltribution and 0:1 1= is the mean of p (or 0-th order equilibrium distribution). Proof: (2) shows that (3) holds for =1.n By induc-tion, assume that (3) holds for any ,n then  and accordingly    1111:112=1:1=022=0:1=011:1=011 =1!1 =!1 =1,1!nn ntxtx nynnyx txllyx nnyx kllnnyxllpy y tnpy knpy yxn   verifies (3) also for==Px ptPt 1.n Further, since 1=1,nP we conclude from (3) th1nnat :1=0!=,lnln . (6) Hence nPx is also given by (4). To prove (5), use 1:1= and (6).□ nnnPxpxExample 1: Ifetrically distributed with X is geom=1xpx and survival function 1=,xPx for x and 0,1 then  1110111 =1=,mmyxmmxy xymPy xy   where the last equality holds true as ,by definition. that any order equilibrium distribution of =1yx=1ymy0my This shows the geometric distribution is identical to the original dis-tribution. Example 2: Let X be a discrete uniform with 1=,px 1m =0,1,2,, .xm 1=01==11nnmnxmxmmn As ,1Copyright © 2011 SciRes. OJDM S. M. LI 129then for ,nm  1=1111=011=111 =11 =,0,1mnnyxnmx nnynnnpxy xmnn ymnmxxmnm    while for 3. Properties of the Equilibrium Distribution Lemma 1 The relationships between raw and factorial >,nm 0.npx In deriving the properties of the higher order equilibrium distributions of p, the following lemmas will be needed. moments are given by nn =1=,and= ,,=1knnkk n knkkSsn (7) where called the first and the secondectively, are given sively by ,,nnkkSs Stirlin=1,2,, ,kng numbers resprecur-11111 1====1, =,withnn nnnkk knSSS SSnS011111 10= 0 ,====1, =,with=0and1.nnnnnn nnnkk knSss s ssksskn Proof: See p.160 in Lemma 2 For and .□ n,y  11=!nyynn1=0 =11!,!11nk kxyknsnnxkssxsnyskss (8)  10=!, 0,1.1nnxnxsxs nss Proof: Let . (9) 1=0=ynxnxIxs It is easy to show that 111=,1nynnsInI yss while s011=.ysIs Then (8) is verified by mathematical induction. To prove (9), simply let in (8). Lemma 3 For and y,ym ,n   11.1!yx y (10) Proof: Since =0xmn!!=ynnmmmnx1=01=1mymxyxm ,then (when and Assuming that it also holds for an arbitrary and then for 10) holds =0n.m =nk ,m=1nk th of (1mee left-hand-sides (LHS)0) beco =0LH =ymxyykmmxy yxkyx yxxmx =0 =0121S = 1!!1!1! =1! 2!1!! 1!!! =11! kkxxkmmkmkxyxmk yxyymk ykmmkkmmkyymkmk =01yxmkmkxmkmkymk22!1 =1.mkmkkm ymk ymk□ Remark: (10) rete version of the formula !12!!1!2!mkis a disc  10111 =.2nm nmynm d=1, 1ynmmnxyx xymn Let 110ˆ=xnnxps spx be the p.g.f. of 1.np The folloxpression for wing theorem gives an e1.ps ˆnTheorem 2  111ˆ1!ˆ=1nnnpsps s1=11111! 1 .!1nnknknkknnnks (11) Proof: Since  1111=1nnyxnnpx pyyx ,  11001111=01ˆ=1 =11 =1xnnxnxxyxnynxyxnps spxnspyyxnpysyx Copyright © 2011 SciRes. OJDM 130 S. M. LI   111=011111111=1111 =111 =1!11! 1 ,by Lemma 2,!1ˆ111! =1 ynyxyxnyynynynnk yknnkknnnnpyssxsnpys nssnyksspsns1=111!1 .!1nknknkknnks□ Theorem 3 :!!=,,!nmnm nnm mnmn , (12) :=1!!=,!mmknm nkknksnmnnk,, (13) hwere :nm is . the -th moment of the distribution Proof:mnp :( )0111=01= =1 =!!! =.!mnm nxynmyxnynnmnxpxnpyxyxnmnmmnTo prove (13), use as stated in Lemma 1, and (12)Consider now the stop-loss transform !1! ,by Lemma 3,nm mnnpyy ::=1=,mmnmk nkks .□ π=xEX x of the r.v. X (where the notation =>0aaIa). For np-loss trand by . Theorem and denote by h stonX (with p) a its n-th fac-formma 1 show ,xsform of nxd LemnEX x the n-tprobability function torial stop-loss transthaEX 1 ant nx=nnEX xP1 and P x =1 kkkk=EX x1.Let nnsnmX ity function al stbe a ra to bfactorindom variable following the probabil- Definenmxe the n-th op-loss transform of mp and nmEX x.mp EX to be the n-th stop-loss transformmhe following theorem holds. Th of Teorem 4 For .p n and, ,mx =!!! =1!nnmmExmnmn Pxmn (14) !!,nmmmnmEXxmnX=1=1!!=!mkkmEX x Emk!! =1!nnkmnnkmkmkkmksmX xksmPxmkn (15) Proof: The argument is similar to that in the above proof.Now define □ =>,,xmmmrx EXxXx  ,m to be the mean residual lifetime (MR and L) of ,mp==mmmPX xhx PXx to be the hazardfunction of Then the following result holds. rate .mp Theorem 5 For x and ,m 11mmPxmPx=1 1mmmrx (16 )  11=.mxrx mh (17) Proof:    111111=>=1 =1 =1 =1,by (14).1mkxmmmmmkx kxmmmmmmkxpkrx EXxXxk xpkpkEX xPxPxmPx  while mPxmPx   1111==111 ==,by (16).1mmmmmmmmmmmpxhx Pxpx PxpxmrxPxPx This proves the conclusion.□ Copyright © 2011 SciRes. OJDM S. M. LI 1314. Equilibrium Distribution and Convolutions section studies the equilibrium distribution of te n-th fold convolution of a counting distribution. The following lemma shows that the usual formulas This h=0=nnknkknxyxk yand 112!nllmn1=11=!!mllnmm,mxxxxxll (18) for ,n also hold for factorial integer powers. Lemma 4 For n   =0=,nnknkknxy xyk   121212=1212ll l nmmlds for =1.n Assume it!=.!! !nmlll mmxx xnxxxlll (19) (18) h holds fo thenand (18) holds by induction. A similar argument proves (19)Next we will discuss the high order equilibrium d-tributions of the convolution of with itself. Let 1 be the n-th fold convo and be the m-th onsider Proof: Clearly or an arbitrary ,n     1=011=0 =011=1 =011=0===11=nnnknkknnknk knkkknknk knkkknknkkxyxyxynnxyxk ynkknnxy xykky xykknxyk           1=nnnx.□ isp **=0=xnnkpxpkp xklution of ,p with *1 =,pporder equilibrium distribution of *nmp*.n Cp*,nk the k-th factorial moment of *,np then  12=1212 =,!!! llnll l knnkll le 12,,,n1!lwher*1=knnkEX X11=1! =!!llnnllknnkEXXll XXX are i.i.d. wiommonth c p.f.whicrecursively by ,p h can be computed  *12 112 1=0*1=0= = =.knnnkklklnnlknklllEXXX XkEX XXEXlkl    The following Theorem gives an expression for *n.mpx Theorem 6 ,m 2,3,n and ,x For   *1*1***1*=1=*1 .nmnnmmnmmnllmlnlmpxpp xmpxl (20) Proof:  1***11*1*1=01*1*=0 11*1*1*0=nmnmxp 1 =1 =11 =mnyxmymnnyxkmxmnnkyxmmnnkxykmnktxmpyyxmpk pykyxmpkpykyxmpkpykyxmpk    1*111*1*10*1*1**=0 11*1 10=0*1*1 1 =1 1 =mnkmnnkx ymnxmnmnnkkxmmmlnmlylnmmpttxkmpkpy ykxmpk pxkpkmpykx yl  *1*11*11=0*1 1*1*11**=01*1 11 =* 1nmnnkxmmlnmllnmmnnmmlnnlmmlkxmpp xpkmkxlmmpp xlpk kx Copyright © 2011 SciRes. OJDM 132 S. M. LI   *1 1*1**=01*111*1*1**=1*11 =* 11 =* .nmmnmnnlmmlnlmlnmmnmnnlmmnllmlmmpp xlpxlmpp xlpx This completes the proof. Remark: Theorem 6 gives a recursive formula for the high order equilibrium distributions of the convolution First obtain □ *.nmplpx and *,kl for om=0,1,,lm starting p.f. and =1,2, ,kn. Then cpute the, followed . xample 3 Conmpx*nmpxEby the cosider nvols ution*2mpx up to X negative binomial ,, for 2 and at is ,1, th10, =1xxpx x for .x Since the ,NB distribution can be viewed as the -th cof a geometric distribution onvolution , =1xgx th can be useden and the above  *=pxg x to compute theorem*=mmg xpx reHere the k-th factorial moment of -cursively. gx is 0=1= !,1kkxkxxk  by Lemma 1. Now 1=xkPx and  *12=1212=121!=!! ! =!111 =!.11!kll llll kklll kkkll lkkk  After some simplifications, we have    11xm*1*1*1 mg xxmm*=*mgx g*1*=1*11 1 =*mmmmllmllmmplgg x  *1 =*1,mzg gxx  z gwhere 1=.1zm This shows that the m-th order ution onomial is a mix-ture of the distributions *1*mggequilibrium distribf a negative bi and ,g where the mixing factor 1=.1zm 5. Equilibrium Distribut i o n o f a M i x t ure ses the equil distribution of a This section discusibriummixed p.f.. For let ,px .xbe the conditional distribution of X, given =. First assume that  has a continuous distribution function U with dens over Then the p.f. of X, given byity u0, . U0=d,px pxequilibriuigh is a Uixture of distributions. The h order m distributions of p are given in the following theorem. Theorem 7 The n-th order equilibrium distribution of the mixed p.f. -m=d,pxpx U0 is given by 0=dnnnpxpx U, (21) where d=nnnEX dUUEX . Proof   =d.nnpx U(1)11100=11 =1dnnyxnnnnnnnnpxpyy xEX xnEXnX xEX UEX EX1n0 =1dnEX x UEX =En   shows n-th. is the mixture of the n-th or-deant i □This theorem that the order equilibrium distribution of a mixed p.fr equilibrium distribution of the conditional p.f., mixed by a new distribution .nU An importxture of geometric p.f.’s, where   10=1d .xpx Uspecial case is the m Geometric mixtures benefit from an important property, is that they Copyright © 2011 SciRes. OJDM S. M. LI 133ar0, for all Then e completely monotone distributions, in the sense that  1nnpx , .xn=npx 1x and ()0d1d01d= =1d =1d ! =,by()1 d,1nnnnxxnnxnxnnnnEX UUEXxUEXUxEXUnEXU   at the n-th order equilibrium distribution of a U-mixture is still a geometric, with same pa-ter. Here the new mixing density is proportional to al one, showing thgeometric ramethe origin .1nnuu Example 4 (Waring Distribution) If =1,xpx for ,x and   111=1,,bau abfor (0, 1), then  1101=, bxa dpx ab ,1=, =,1xababbab xaaxab  Waring distribution. which is called a It follows that  11= npxwhich is a Then =1,1,bnxannpxu  distribution when ,anbn >.bn  10=1 d,1 px =1bn abxanan xab  when ,bn then  does not exist. npxIf instead, the geometric p.f. is mixed over another discrete p.f., that ,is =1=1kxjjf jjpx for =,xnnUxanbnanbn  ,x1, where 0< j =1kjfj =1, then ,where  =1,kxnnjjpxfjx =1j=11=,1njjnnkiiifjfjfi for =1,2,, .jk The following theorem gives the aging properties of the higher order equilibrium distributions of geometric mixtures. um distribution of Theorem 8 The n-th order equilibria gure is DFR and eometric mixtd.dIMRL Also nPx ,Px for 1nequivalently, 0,x or 1<.nstnpp ixture, it isProof: Sincege completely monotone. Then is see . Further, p nis also a ometric mnpdDFR ,1=1 ,nnrxh x by (1nce is 7), henp.dIMRL Lastly, 101=11nnnnPx rPx rx, by (16). 6. Equilibrium Distribution of the (a,b) Family nside tions of the □ We cor here the equilibrium distribu,b class of discrete distributions, or more precisely, the important subclass of the ,ab famthe aof ily called,,0ab class, see [11 dis-the non-negative integers h the re,12]. Thountingtributions has support on onwhiccurrence relationis class of c =1pxabx px holds for . The members of the this class are binomial, Poisson and negative binomial distributions (with their corresponding special cases). It is easily seen that =1,2,x 11=and= ,kfor2.11kak bab kaa Then we have the following recursive formula for .np Theorem 9 The n-th order equilibrium distribution np in the ,,0ab class of distributions satisfies the following recursion for n: Copyright © 2011 SciRes. OJDM 134 S. M. LI    11 ,.nnana b axpx xnanab  The starting poinursion are 111 1=1nnnnaxnap xnanab  11p xa (22)ts of the rec px111=,px Px 1110= 10pp and   11=10= 110!nnnkknppk11!,nkn r Profo 2.n of:  =1,bpxapxx or equivalently, =1 1,x pxaxpx for =0,1,2, .x Then abpx   1=2 nnyp y yxay n1nn111211 11 yxnnyxyxyxyxabaypyyxxpyyxax n111=1,1=111nyxnnpy yxnnypy yx abpxnn       121111111111=221 1=1111 ,1nnyxnnyxnnyxnnyxnnnnnnnnnpy yxnpyyxnapyyxnx napyyxnpxxnpxnnapxaxnpxn in turns implies 1 nxnpy yx which  111111= 11 ,1nnnnnnnnnpx apxpxnnan ab axpxnn  11=,nnaan a b Since we get (22). Finally, we have    11111=011=0 1111=10= 11 =11! =!1! =1102.!nnynnnk kyknnknkkynnknnkknnppyynnpy yknpyyknpIn k This completes the proof. For another subclass of the family, the □ ,ab,,1ab class of distributions, ttion he rela=px 1abxpx holds forere x2, wh0p is an arbitrarily selected value in 0,1. In this easy to see that case, it is  1=,1kkak ba for The above recursive formula (22) and those for the starting points still hold true here, the only change being that 2.k111=.1pabpa  07. Conclusions This paper investigates the higher order equilibrium dis-tributions of counting random variables. The above re-sults can be used in Risk Theory to derive bounds of ruin probabilities in the discrete time risk model. They also leade factorial momentated random va the surplus before t at ruin and the ti Garrido  for details). 8. References  E. Fagiuoli and F. Pellerey, “New Partial Orderings and Applications,” Naval Research Logistics, Vol. 40, No. 6, 1993, pp. 829-842. to ths of three relriables:ruin, the deficime of ruin (see Li anddoi:10.1002/1520-6750(199310)40:6<829::AID-NAV3220400607>3.0.CO;2-D  E. Fagiuoli and F. Pellerey, “Preservation of Certain Copyright © 2011 SciRes. OJDM S. M. LI Copyright © 2011 SciRes. OJDM 135Classes of Life Distributions under Poisson Shock Mod-els,” Journal of Applied Probability, Vol. 31, No. 2, 1994, pp. 458-465. doi:10.2307/3215038  A.K. Nanda, H. Jain and H. Singh, “On Closure of SomePartial Orderings under Mixture,” Journal of AppliedProbability, Vol. 33, No. 3, 1996, pp. 698-706. doi:10.2307/3215351  O. Hesselager, S. Wang and G.E. Willmot, “Exponential and Scale Mixture and Equilibrium Distributions,” Scan-dinavian Actuarial Journal, Vol. 20, No. 2, 1994, pp. 125-142.  Kickman, D. Jones and C.atics,” 2nd Edition, Socie X. Lin ve Renewal Equationuin TVol. M. Bowers, H. Gerber, J. Nesbitt, “Actuarial Mathem ty of Actuaries, Schaumburg, 1997. and G. E. Willmot, “Analysis of a Defecti- Arising in Rheory,” Insurance: Mathematics and Economics, 25, No. 1, 1999, pp. 63-84. doi:10.1016/S0167-6687(99)00026-8  X. Lin and G. E.ents ofime of Willmot, “The Mom the TRuin, the Surplus before Ruin, and the Deficit at Ruin,” Insurance: Mathematics and Economics, Vol. 27, No. 1, 2000, pp. 19-44. doi:10.1016/S0167-6687(00)00038-X  G. E. Willmot, “Bounds for Compound Distributions Based on Mean Residual Lifetimes and Equilibrium Dis-tributions,” Insurance: Mathematics and Economics, Vol. 21, No. 1, 1997, pp. 25-42. doi:10.1016/S0167-6687(97)00016-4  G. E. Willmot and J. Cai, “Aging and other DistProperties of Discreteributional Compound Geometric Distribu-tions,” Insurance: Mathematics and Economics, Vol. 28, No. 3, 2001, pp. 361-379. doi:10.1016/S0167-6687(01)00062-2  R. W. Hamming, “Numerical methods for scientists and Bulletin, Vol. 12, No. 1, Engineers,” 2nd Edition, Dover, New-York, 1973.  H. H. Panjer, “Recursive Evaluation of a Family of Com-pound Distributions,” ASTIN 1981, pp. 22-26.  S. A. Klugman, H. H. Panjer and G. E. Willmot, “Loss Models,” Wiley, New-York, 1998.  S. Li and J. Garrido, “On the Time Value Ruin of Dis-crete Time Risk Process,” Working Paper 02-18, Univer-sidad Carlos III of Madrid, Madrid, 2002.