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.
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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
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
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)  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) .
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) , and Feller (1968) ) 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 
and ). 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)  we deduce that and where and and 03BB1,2 are given by (see also , ,  and ) . 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
, Hardin and Sweet  and Blasi .
. 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)  for the particular case r 0
(see also ).
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  BLASI (A.) .2014 On a random walk between
Ann. Probability, 4 (1976) pp. 695-696.  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,
(third edition) Vol. 1, John Wiley, New York (1968).
A note on absorption probabilities for
 HARDIN (J.C.) and SWEET (A.L.) .  FELLER 2014 2014 random walk between a reflecting and
J. Appl. Prob., 6 (1969) pp. 224-226.  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.
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MUNFORD (A.G.) .
J. R. Statist. Soc., B 43 (1981) pp. 142-146.  MURTHY (K.P.N.) and KEHR (K.W.).2014 Mean
walks on a random lattice,
Phys. Rev., A 40, N4 (1989) pp. 2082-2087.  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
procedures and single-server queues,
J. Appl. Prob., 24 (1987) pp. 200-214.
The random walk between a reflecting and
 WEESAKUL (B.) .  a a reflecting to cumulative sum 2014 barrier,
Ann. Math. Statist., 32 (1961) pp. - 765-769. 103 2014 an absorbing ...
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