ON THE DISTRIBUTION OF THE MAXIMUM OF SUMS OF A SEQUENCE OF INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES.

Transcription

1 TWMS J. Appl. Eng. Math. V.6,N.1, 2016 pp ON THE DISTRIBUTION OF THE MAXIMUM OF SUMS OF A SEQUENCE OF INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES. T.M.ALIYEV 1, SH. B. BAKHSHIYEV 1, Abstract. 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. Keywords: random walk, Laplace transform, generating function. AMS Subject Classifiction: 60A10, 60J25, 60G05, 60G Introduction In the following paragraphs, we consider a sequence of independent and identically distributed random variables ξ 1, ξ 2,..., ξ n,... given on the probability space Ω, F, P ( ), with ξ i 1, 0, 1, 2,..., i 1 and P ξ i = k = P k, i 1, P 2 = P 3 =... = 0. In an integer phase it is natural to call such processes downwardly continuous [1]. From the sums S 0 = 0, S n = ξ 1 + ξ ξ n, n 1, we obtain the sequence S n, n 1, which constitutes a random walk. µ n = max 0 i n S i, n = 0, 1, 2,.... Our aim is to find the distribution of the variable µ n, n 0 for such a random walk. Problems of this kind arise while investigating queue systems, inventory management systems, etc. Many authors studied a number of problems of theory of random walks in the plane. Recently, special emphasis is placed on the investigation of random walks in the subexponential case and two-boundary problems for them. For instance, D. A. Korshunov considers in his paper [2] the random walk S n, n 1 with the negative drift Mξ 1 = m < 0 and an excess with heavy tails, when Me λξ 1 = for any λ > 0. The author investigated only the asymptotic behavior at infinity of the distribution density of the supremum sup S n. I. I. Yezhov, V. F. Kadankov, T. V. Kadankova [3] consider the sequence of independent and identically distributed random variables ξ R = (, + ), P ξ = 0 < 1 and ξ n, n N = 1, 2, 3, Institute of Control Systems. Azerbaijan National Academy of Sciences, 9. B. Vahabzade Str. Az 1141, Baku, Azerbaijan. tofik aliev44@rambler.ru; shaxbazia@yahoo.com; Manuscript received: January 29, 2015; Accepted: June 15, TWMS Journal of Applied and Engineering Mathematics, Vol.6, No.1; c Işık University, Department of Mathematics, 2016; all rights reserved. 70 n

2 T.M.ALIYEV, SH. B. BAKHSHIYEV: ON THE DISTRIBUTION OF THE MAXIMUM Considering the random walk ξ n R, n Z = 0, 1,... generated by the random variable ξ = ξ(0) = 0, ξ n = ξ ξ n, n N for all x 0, they investigated the random variables t x = infn : ξ(n) > x, T x = ξ(t x ) x, t x = infn : ξ(n) < x, T x = ξ(t x ) x. Using generating functions of the joint distributions t x, T x and t x, T x for the right half-plane R(p) 0 (p is the parameter), in terms of generating functions and Laplace transform, the joint distribution of the first exit time from the interval and the value of the overshoot by the random walk was determined. However, practically, this solution method for two-boundary problems for random walks is unacceptable for special class problems, since it is based on the use of distribution of one-boundary functionals, whereas for determining the latter, combinatorial and factorization methods are used. In the special class problems, using methods of the theory of generating functions and principles of random walks, we consider the problem of determining the distribution of the variable µ n, n 0, which, in turn, allows us to determine the distribution ρ n (k) = P S n = k of the random walk S n, n 0. It is important that to do that we have to deal only with the distribution P µ n = 0, n 0. Let us proceed to determine the distribution of the random variable µ n, n 0. It stands to reason that the generating function ϕ(z) of the random variable ξ 1, i.e. ϕ(z) = Mz ξ 1 = P k z k, z 1 k= 1 is not an analytic function at the point 0. And if we introduce the new variable its generating function ϕ (z) = zϕ(z) = is an analytic function in the circle z 1. For S n, n 1, we get then Seeing that ξ i = ξ 1 + 1, i 1, k= 1 π k z k, π k = P k 1 Mz Sn = Mz (ξ ξ n) z n = [ϕ (z)] n z n. ϕ (z) = zϕ(z) = [ϕ (z)] n = k=0 π k z k, z 1, k=0 π n k zk, z 1, where π n k is n-fold convolution of distribution, π k, k 1. According to the principles of the random walk theory [4], the following holds: (1) π 2 k = π 0π k + π 1 π k π k π 0 π 3 k = π2 0 π k + π1 2 π k πk 2 π

3 72 TWMS J. APPL. ENG. MATH, V.6, N.1, 2016 Thus, the sequence πk n is known for any k, n Main results With the known sequence (1), let us proceed to determine the explicit form of the generating function from the variables x n = P µ n = 0. It should be noted that it follows from the formation of the sequence πk n that P S n = k = πk+1 n = ρ n(k) is the probability of the event S n = k. Assume that we reach the trajectory peak of the walk S n, n 1, i.e. µ i k. Two options are possible here: either the endpoint is below the level k, then S n k, or neither upper nor lower events cross on the level k - then S n < k. In that case, µ k. Therefore, the following equality holds: (2) P µ n k = P S n k + P µ k, S n < k. µ i being the maximum among all peaks of our walk, at least one such i exists when S i k. If it turns out that S i = k, there will be such instant j, when S j = k. Since at some point we are above the level k, we cannot pass it getting below the level k, because we progress only by unity. If at that last instant S j = k, 0 j n 1, then we are below the level k at all subsequent instants. (3) C j = S j = k, S j+1 < k,..., S n < k. Then the event µ k, S n < k is the union of events C j, i.e. µ k, S n < k = By virtue of (2) and (3), we have the following: P µ n k = P S n k + P S j = k, S j+1 < k,..., S n < k. j=o In what follows (4) P S j = k, S j+1 < k,..., S n < k = P S j = kp S j+1 < k,..., S n < k/s j = k = = ρ j (k)p ξ ξ j+1 < k,..., ξ ξ n < k/ξ ξ j = k = = ρ j (k)p ξ j+1 < 0,..., ξ j ξ n < 0/ξ ξ j = k = C j. = ρ j (k)p ξ j+1 < 0,..., ξ j ξ n < 0. Since ξ 1, ξ 2,... are independent and identically distributed random variables, (5) S 1 < 0, S 2 < 0,..., S n j < 0 ξ 1 < 0, ξ 1 + ξ 2 < 0,... ξ 1 + ξ ξ n j < 0, i.e. these two events are identically equal. Considering that ξ 1 = 1, it follows from the right-hand side of (5) that ξ 2 0; and from the previous inequality, we get that ξ ξ n j 0. By virtue of this reasoning from (4) we obtain (6) P S j = k, S j+1 < k,..., S n < k = ρ j (k) P ξ 0 = 1, S 1 0,..., S n j 1 0 = = ρ j (k) π 0 P S 1 0,..., S n j 1 0 = ρ j (k) π 0 P max S i 0. 1 i n j 1

4 T.M.ALIYEV, SH. B. BAKHSHIYEV: ON THE DISTRIBUTION OF THE MAXIMUM By convention, S 0 = 0. Hence, it follows from (6) that P S j = k, S j+1 < k,..., S n < k = ρ j (k) π 0 P max S i 0 = 1 i n j 1 = ρ j (k) π 0 P max S i = 0. 0 i n j 1 then (7) (8) Thus, Since By virtue of (7) We get If and P µ n k = P S n k + π 0 ρ j (k) P µ n j 1 = 0. P µ n k = j k P S n k = j k ρ n (j), ρ n (j) + π 0 ρ j (k) P µ n j 1 = 0. P µ n = 0 = P S n = 0 + π 0 [ρ j (1) ρ j (1)] P µ n j 1 = 0. x n = P µ n = 0. x n = ρ n (0) + π 0 [ρ j (0) ρ j (1)] x n j 1. x(z) = ρ k (z) = x n z n, z 1 n=0 ρ n (k) z n, z 1, n=0 then the equation (8) in convolutions takes on the following form: hence x(z) = ρ 0 (z) + π 0 (z) [ ρ 0 (z) ρ 1 (z)], x(z) = ρ 0 (z) 1 π 0 (z) [ ρ 0 (z) ρ 1 (z)]. Knowledge of the generating function x(z) allows for determining the probabilities of the event S n = k. ρ n (k) = P S n = k = π n k+n

5 74 TWMS J. APPL. ENG. MATH, V.6, N.1, 2016 References [1] Nelson,R., (1995), Probability, stochastic processes and queening theory: the mathematics of computer performance modeling, Springer- Verlag, Inc. New York, pp [2] Korshunov,DA., (2006), On the distribution density of the supremum of a random walk in the subexponential case. Siberian Mathematical Journal, Vol.74, No.6, pp (in Russian) [3] Yezhov,I.I., Kadankov,V.F. and Kadankova,T.V., (2007), Main two-boundary problems for a random walk. ISSN Ukrainian Mathematical Journal, Vol.59, No.11, pp (in Russian) [4] Nasirova,T. I. and Aliyev,T.M., (2011), Probability. Monograph (in Azerbaijani). Baku, pp Tofiq Aliyev, for a photograph and biography, see TWMS Journal of Applied and Engineering Mathematics, Volume 4, No.2, 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 of Sciences (ANAS), Azerbaijan. Since 2000 he is researcher at the laboratory Department of Economo and mathematical methods at the Institute of Control Systems of Azerbaijan National Academy of Sciences. His research interests are economic methods, queuing theory.