The authors find in terms of generating functions the distribution of the maximum of sums of independent and identically distributed random variables with no negative jumps different from −1.Publisher's Versio

Aliyev, Tofig M.

Bakshiev, Shahin B.

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. Appl. Eng. Math. V.6 ,N.1, 2016 pp. 70-74ON THE DISTRIBUTION OF THE MAXIMUM OF SUMS OF ASEQUENCE OF INDEPENDENT AND IDENTICALLY DISTRIBUTEDRANDOM VARIABLES.T.M.ALIYEV1, SH. B. BAKHSHIYEV 1, §Abstract. The authors find in terms of generating functions the distribution of themaximum of sums of independent and identically distributed random variables with nonegative jumps different from −1.Keywords: random walk, Laplace transform, generating function.AMS Subject Classifiction: 60A10, 60J25, 60G05, 60G10.1. IntroductionIn the following paragraphs, we consider a sequence of independent and identically dis-tributed random variables {ξ1, ξ2, . . . , ξn, . . .} given on the probability space {Ω, F, P (·)},with ξi ∈ {−1, 0, 1, 2, . . .}, i ≥ 1 and P{ξi = k} = Pk, i ≥ 1 , P−2 = P−3 = . . . = 0.In an integer phase it is natural to call such processes downwardly continuous [1].From the sums S0 = 0, Sn = ξ1 + ξ2 + . . . + ξn, n ≥ 1, we obtain the sequence {Sn},n ≥ 1, which constitutes a random walk.Let us denoteµn = max0≤i≤nSi, n = 0, 1, 2, . . . .Our aim is to find the distribution of the variable µn, n ≥ 0 for such a random walk.Problems of this kind arise while investigating queue systems, inventory managementsystems, etc.Many authors studied a number of problems of theory of random walks in the plane. Re-cently, special emphasis is placed on the investigation of random walks in the subexponen-tial case and two-boundary problems for them. For instance, D. A. Korshunov considersin his paper [2] the random walk {Sn}, n ≥ 1 with the negative drift Mξ1 = −m < 0 andan excess with heavy tails, when Meλξ1 =∞ for any λ > 0. The author investigated onlythe asymptotic behavior at infinity of the distribution density of the supremum supnSn.I. I. Yezhov, V. F. Kadankov, T. V. Kadankova [3] consider the sequence of independentand identically distributed random variables ξ ∈ R = (−∞,+∞), P{ξ = 0} < 1 and ξn,n ∈ N = {1, 2, 3, . . .}.1 Institute of Control Systems. Azerbaijan National Academy of Sciences, 9. B. Vahabzade Str. Az1141, Baku, Azerbaijan.e-mail: tofik aliev44@rambler.ru; shaxbazia@yahoo.com;§ Manuscript received: January 29, 2015; Accepted: June 15, 2015.TWMS Journal of Applied and Engineering Mathematics, Vol.6, No.1; c© Işık University, Departmentof Mathematics, 2016; all rights reserved.70T.M.ALIYEV, SH. B. BAKHSHIYEV: ON THE DISTRIBUTION OF THE MAXIMUM... 71Considering the random walk ξn ∈ R, n ∈ Z = {0, 1, . . .} generated by the randomvariable ξ = ξ(0) = 0, ξn = ξ1 + . . . + ξn, n ∈ N for all x ≥ 0, they investigated therandom variablestx = inf{n : ξ(n) > x}, T x = ξ(tx)− x,tx = inf{n : ξ(n) < −x}, Tx = −ξ(tx)− x.Using generating functions of the joint distributions {tx, T x} and {tx, Tx} for the righthalf-plane R(p) ≥ 0 (p is the parameter), in terms of generating functions and Laplacetransform, the joint distribution of the first exit time from the interval and the value ofthe overshoot by the random walk was determined. However, practically, this solutionmethod for two-boundary problems for random walks is unacceptable for special classproblems, since it is based on the use of distribution of one-boundary functionals, whereasfor determining the latter, combinatorial and factorization methods are used.In the special class problems, using methods of the theory of generating functions andprinciples of random walks, we consider the problem of determining the distribution ofthe variable µn, n ≥ 0, which, in turn, allows us to determine the distribution ρn(k) =P{Sn = k} of the random walk {Sn}, n ≥ 0. It is important that to do that we have todeal only with the distribution P{µn = 0}, n ≥ 0.Let us proceed to determine the distribution of the random variable µn, n ≥ 0. It standsto reason that the generating function ϕ(z) of the random variable ξ1, i.e.ϕ(z) = Mzξ1 =∞∑k=−1Pkzk, |z| ≤ 1is not an analytic function at the point 0. And if we introduce the new variableξ∗i = ξ1 + 1, i ≥ 1,its generating functionϕ∗(z) = zϕ(z) =∞∑k=−1πk zk, πk = Pk−1is an analytic function in the circle |z| ≤ 1.For Sn, n ≥ 1, we getMzSn = Mz(ξ∗1+...ξ∗n)zn =[ϕ∗(z)]nzn.Seeing thatϕ∗(z) = zϕ(z) =∞∑k=0πk zk, |z| ≤ 1,then[ϕ∗(z)]n =∞∑k=0πn∗k zk, |z| ≤ 1,where πn∗k is n-fold convolution of distribution, πk, k ≥ 1.According to the principles of the random walk theory [4], the following holds:π2∗k = π0πk + π1πk−1 + . . .+ πkπ0π3∗k = π2∗0 πk + π2∗1 πk−1 + . . .+ π2∗k π0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1)72 TWMS J. APPL. ENG. MATH, V.6, N.1, 2016Thus, the sequence{πn∗k}is known for any k, n ≥ 1.2. Main resultsWith the known sequence (1), let us proceed to determine the explicit form of thegenerating function from the variables xn = P{µn = 0}. It should be noted that itfollows from the formation of the sequence {πn∗k } that P{Sn = k} = πn∗k+1 = ρn(k) is theprobability of the event {Sn = k}.Assume that we reach the trajectory peak of the walk {Sn}, n ≥ 1, i.e. µi ≥ k. Twooptions are possible here: either the endpoint is below the level k, then Sn ≥ k, or neitherupper nor lower events cross on the level k - then Sn < k. In that case, µn−1 ≥ k.Therefore, the following equality holds:P{µn ≥ k} = P{Sn ≥ k}+ P{µn−1 ≥ k, Sn < k}.(2)µi being the maximum among all peaks of our walk, at least one such i exists whenSi ≥ k. If it turns out that Si = k, there will be such instant j, when Sj = k. Since atsome point we are above the level k, we cannot pass it getting below the level k, becausewe progress only by unity. If at that last instant Sj = k, 0 ≤ j ≤ n− 1, then we are belowthe level k at all subsequent instants.Let us denoteCj = {Sj = k, Sj+1 < k, . . . , Sn < k}.Then the event {µn−1 ≥ k, Sn < k} is the union of events Cj , i.e.{µn−1 ≥ k, Sn < k} =n−1⋃j=0Cj .(3)By virtue of (2) and (3), we have the following:P{µn ≥ k} = P{Sn ≥ k}+n−1∑j=oP{Sj = k, Sj+1 < k, . . . , Sn < k}.In what followsP{Sj = k, Sj+1 < k, . . . , Sn < k} = P{Sj = k}P{Sj+1 < k, . . . , Sn < k/Sj = k} == ρj(k)P{ξ1 + . . .+ ξj+1 < k, . . . , ξ1 + . . .+ ξn < k/ξ1 + . . .+ ξj = k} == ρj(k)P{ξj+1 < 0, . . . , ξj+1 + . . .+ ξn < 0/ξ1 + . . .+ ξj = k} == ρj(k)P{ξj+1 < 0, . . . , ξj+1 + . . .+ ξn < 0}.(4)Since ξ1, ξ2, . . . are independent and identically distributed random variables,{S1 < 0, S2 < 0, . . . , Sn−j < 0} ≡ {ξ1 < 0, ξ1 + ξ2 < 0, . . . ξ1 + ξ2 + . . .+ ξn−j < 0},(5)i.e. these two events are identically equal. Considering that ξ1 = −1, it follows fromthe right-hand side of (5) that ξ2 ≤ 0; and from the previous inequality, we get thatξ2 + . . .+ ξn−j ≤ 0.By virtue of this reasoning from (4) we obtainP{Sj = k, Sj+1 < k, . . . , Sn < k} = ρj(k) P{ξ0 = −1, S1 ≤ 0, . . . , Sn−j−1 ≤ 0} == ρj(k) π0 P{S1 ≤ 0, . . . , Sn−j−1 ≤ 0} = ρj(k) π0 P{max1≤i≤n−j−1Si ≤ 0}.(6)T.M.ALIYEV, SH. B. BAKHSHIYEV: ON THE DISTRIBUTION OF THE MAXIMUM... 73By convention, S0 = 0. Hence, it follows from (6) thatP{Sj = k, Sj+1 < k, . . . , Sn < k} = ρj(k) π0 P{max1≤i≤n−j−1Si ≤ 0}== ρj(k) π0 P{max0≤i≤n−j−1Si = 0}.Thus,P{µn ≥ k} = P{Sn ≥ k}+ π0n−1∑j=0ρj(k) P {µn−j−1 = 0} .SinceP{Sn ≥ k} =∑j≥kρn(j),thenP{µn ≥ k} =∑j≥kρn(j) + π0n−1∑j=0ρj(k) P {µn−j−1 = 0} .(7)By virtue of (7)P{µn = 0} = P{Sn = 0}+ π0n−1∑j=0[ρj(1)− ρj(1)]P{µn−j−1 = 0}.Let us denotexn = P{µn = 0}.We getxn = ρn(0) + π0n−1∑j=0[ρj(0)− ρj(1)]xn−j−1.(8)Ifx(z) =∞∑n=0xn zn, |z| ≤ 1andρ̃k(z) =∞∑n=0ρn(k) zn, |z| ≤ 1,then the equation (8) in convolutions takes on the following form:x(z) = ρ̃0(z) + π0(z) [ρ̃0(z)− ρ̃1(z)] ,hencex(z) =ρ̃0(z)1− π0(z) [ρ̃0(z)− ρ̃1(z)].Knowledge of the generating function x(z) allows for determining the probabilitiesρn(k) = P{Sn = k} = πn∗k+nof the event {Sn = k}.74 TWMS J. APPL. ENG. MATH, V.6, N.1, 2016References[1] Nelson,R., (1995), Probability, stochastic processes and queening theory: the mathematics of computerperformance modeling, Springer- Verlag, Inc. New York, pp. 583.[2] Korshunov,DA., (2006), On the distribution density of the supremum of a random walk in the subex-ponential case. Siberian Mathematical Journal, Vol.74, No.6, pp. 1296-1302. (in Russian)[3] Yezhov,I.I., Kadankov,V.F. and Kadankova,T.V., (2007), Main two-boundary problems for a randomwalk. ISSN 1027-3190. Ukrainian Mathematical Journal, Vol.59, No.11, pp. 1485-1509. (in Russian)[4] Nasirova,T. I. and Aliyev,T.M., (2011), Probability. Monograph (in Azerbaijani). Baku, pp. 178.Tofiq Aliyev, for a photograph and biography, see TWMS Journal of Applied and Engineering Mathe-matics, Volume 4, No.2, 2014.Shahin B. Bakshiev was born in Azerbaijan Republic, in Baku. He received his M.S.degree Azerbaijan Technical University and his Phd from Azerbaijan National Academy ofSciences (ANAS), Azerbaijan. Since 2000 he is researcher at the laboratory Department ofEconomo and mathematical methods at the Institute of Control Systems of AzerbaijanNational Academy of Sciences. His research interests are economic methods, queuingtheory.

On the distribution of the maximum of sums of a sequence of independent and identically distributed random variables

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On the distribution of the maximum of sums of a sequence of independent and identically distributed random variables

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