AFST_1992_6_1_1_95_0

AFST_1992_6_1_1_95_0 - A NNALES DE LA FACULTÉ DES SCIENCES...

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Unformatted text preview: A NNALES DE LA FACULTÉ DES SCIENCES DE T OULOUSE A HMED E L -S HEHAWY On absorption probabilities for a random walk between two different barriers Annales de la faculté des sciences de Toulouse 6e série, tome 1, no 1 (1992), p. 95-103. <http://www.numdam.org/item?id=AFST_1992_6_1_1_95_0> © Université Paul Sabatier, 1992, tous droits réservés. L’accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/∼annales/), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Serie 6, Vol. I, n° 1, 1992 Annales de la Faculte des Sciences de Toulouse On for a random walk between two different barriers absorption probabilities EL-SHEHAWY M. AHMED(1) On determine les probabilités d’absorption de la marche aléatoire stationnaire la plus générale sur les entiers ~0, 1, ... , N~, ou l’origine (site N) est une barriere partiellement réfléchissante et le site N (site 0) est absorbant. Nous donnons des expressions du temps d’absorption. explicites, de 1’esperance et de la variance ABSTRACT. - A determination is made of the absorption probabilities of the most general stationary random walk on the integers ~0, 1, ... , N~ where by the origin (site N) is a partially reflecting barrier and the site N (site 0) is an absorbing one. Explicit expressions are given for the mean and the variance of the time to absorption. KEY-WORDS : Absorption time; Absorption probability; generating function; partial fractions. 1. Introduction a random walk on a line-segment of N + 1 sites as shown in and b). The N + 1 sites on the line-segment are denoted by figure a) the integers 0, 1, 2, ..., N. Let p be the probability for a particle (per unit time) to move from a site j, 0 j N, to its nearest neighbor on the right, j + 1, and q be the probability to move from j to j - 1. The probability to stay (for one unit time) at a site j is thus r = 1 - p - q. There are two Consider 1 (~) Department of Math., Damietta Faculty of Science, Damietta, - 95 - Egypt one of which, site N (site 0), is absorbing and the other, site 0 is partially-reflecting, that is the probability to stay at a site 0 (site (site N) N) is a and the probability to reflect to a site 1 (site N - 1) is/3= 1 - a, barriers, respectively. Fig. arrows 1 A moving particle and stayed at the same on a site line segment: right and left jumps are indicated by by loops. The two situations a) and b) in figure 1 can be easily obtained from each other by replacing j with N - j and interchanging p and q, respectively. So we deal with the second one of them. Let gjo(t) be the probability that the particle is absorbed at 0 at time given that its initial site was j. Weesakul (1961) [11] has computed the probability gjo(t), in the special cas r 0, a p; however, his calculations t = contained some The probability special errors. gjo(t) two cases = Correct formulae can be found in Blasi (1976) [1]. is also given by Hardin and Sweet (1969) ~4~ in the and In most text books covering random walks (for example Cox and Miller (1965) [2], and Feller (1968) [3]) the determination of explicit expressions for the absorption probabilities from the generating function is effected by partial fractions; however, it has generally been difficult to obtain (see [4] and [7]). In this note, determination of generalization expression for gjo(t) from the corresponding generating function by partial fraction expansions is presented. This generalization expression apparently is not covered by the literature. Explicit formulae for the mean and the variance of the time to absorption are also given. 2. Partial fraction expansion The probability gjo(t) that the particle is at location 0 for the first time after t steps given that its initial position was j obeys the following difference equation: We set For j = N Following we have Neuts (1963) [9] we deduce that and where and and 03BB1,2 are given by (see also [4], [5], [6] and [8]) . Both the numerator and the denominator of equation (3) have degree N. If the roots of To(z), ZI, z2, ..., ZN are distinct, (3) can be decomposed into partial fractions as where a’s can be determined by In order to determine the roots of the denominator formation In terms of the transformation and (3) study (8), Uo(w) is given by of the function shows that denominator N make the trans- becomes where the denominator A we ~/ [(o: - Uo(w) has N distinct roots The roots are then 2q + /3]. wk, k . . = 1, 2, ..., N if q + /3L ~/ ~~a - r) q/p - 2distinctthere roots If N > k = 2, 3, ..., N, that is given by where w1 is the From and (8) we unique give root of the are only z~ of N - 1 distinct roots w~, The remaining root equation have so From is (6) we can obtain the coefficient of zt in the Explicit generalization expression finally becomes where which can be rewritten as: for the expansion of Gj(z) which absorption probability gjo(t) where and zl with the smallest root in absolute value of Using the following theorem ( ~3~, p. 277) To(z). with appropriate change of notation THEOREM . If Gj(z) is a rational function with a simple root zl of the denominator which is smaller in absolute value than all other roots, then the coefficient gj0(t) of zt is given asymptotically by where al is We where find defined in r17~. that is defined previously. We see that with the appropriate change of notation in the special cases considered in the introduction, formula (20) agrees with that of Weesakul [11], Hardin and Sweet [4] and Blasi [1]. . 3. The Let Ej starts at a mean and the variance of the process denotes the time up to absorption at site j, 0 j Nthen a site 0 when the particle and hence Using (3) When p = and q, (22), we get limp-q p is evaluated using 1’Hospital’s rule, This result was established by Khan with appropriate change of notation and in this case ( 1984) [6] for the particular case r 0 (see also [10]). The variance of the absorption time Ej can be obtained from the following relation in the form = where and When q = p, found to be limq_,p Var(Tj) is evaluated using I’Hospital’s rule, and it is where References [1] BLASI (A.) .2014 On a random walk between Ann. Probability, 4 (1976) pp. 695-696. [2] Cox (D.R.) and MILLER Methuen, London (H.D.) . (1965). 2014 The a reflecting and absorbing barrier, theory of stochastic processes, An Introduction to probability theory and its applications, (W.) . (third edition) Vol. 1, John Wiley, New York (1968). A note on absorption probabilities for [4] HARDIN (J.C.) and SWEET (A.L.) . [3] FELLER 2014 2014 random walk between a reflecting and J. Appl. Prob., 6 (1969) pp. 224-226. [5] KENNEDY server (D.P.) . Appl., Some martingales related to cumulative sum tests and single4 (1976) pp. 261-269. KHAN (R.A.) . On cumulative sum procedures and the SPRT J. R. Statist. Soc., B 64 (1984) pp. 79-85. 2014 a absorbing barrier, queues, Stoch. Proc. [6] 2014 an applications, random walk with [7] A first passage problem in MUNFORD (A.G.) . control application, J. R. Statist. Soc., B 43 (1981) pp. 142-146. [8] MURTHY (K.P.N.) and KEHR (K.W.).2014 Mean walks on a random lattice, Phys. Rev., A 40, N4 (1989) pp. 2082-2087. [9] NEUTS (M.F.) . Absorption probabilities for a random walk between and an absorbing barrier, Bull. Soc. Math. Belgique, 15 (1963) pp. 253-258. 2014 a first-passage 2014 STADJE 2014 quality time of random Asymptotic behaviour of a stopping time related (W.) . procedures and single-server queues, J. Appl. Prob., 24 (1987) pp. 200-214. The random walk between a reflecting and [11] WEESAKUL (B.) . [10] a a reflecting to cumulative sum 2014 barrier, Ann. Math. Statist., 32 (1961) pp. - 765-769. 103 2014 an absorbing ...
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