A note on random walks

1971 ◽  
Vol 8 (01) ◽  
pp. 198-201 ◽  
Author(s):  
R. M. Phatarfod ◽  
T. P. Speed ◽  
A. M. Walker
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variables ◽  
Image Position ◽  
Mutually Independent ◽  
Reflecting Barriers ◽  
Independent And Identically Distributed

Let {Xn } be a random walk between reflecting barriers at 0 and a > 0 with jumps {Zn }. By we mean the random walk between absorbing barriers at — a and 0+ with the same jumps {Zn }. It has been known for some time that when {Zn } is a sequence of mutually independent and identically distributed random variables, and 0 ≦x <a, we have for all n:

A note on random walks

1971 ◽  
Vol 8 (1) ◽  
pp. 198-201 ◽  
Author(s):  
R. M. Phatarfod ◽  
T. P. Speed ◽  
A. M. Walker
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variables ◽  
Image Position ◽  
Mutually Independent ◽  
Reflecting Barriers ◽  
Independent And Identically Distributed

Let {Xn} be a random walk between reflecting barriers at 0 and a > 0 with jumps {Zn}. By we mean the random walk between absorbing barriers at — a and 0+ with the same jumps {Zn}. It has been known for some time that when {Zn} is a sequence of mutually independent and identically distributed random variables, and 0 ≦x <a, we have for all n:

Poisson recurrence times

1967 ◽  
Vol 4 (03) ◽  
pp. 605-608
Author(s):  
Meyer Dwass
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variables ◽  
Recurrent Event ◽  
Recurrence Times ◽  
Image Position ◽  
Independent And Identically Distributed

Let Y1, Y 2, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let Sn = Y 1 + … + Yn, n = 1,2,…, S 0 = 0. The event Sn = n is a recurrent event in the sense that successive waiting times between recurrences form a sequence of independent and identically distributed random variables. Specifically, the waiting time probabilities are (Alternately, the fn can be described as the probabilities for first return to the origin of the random walk whose successive steps are Y1 − 1, Y2 − 1, ….)

Poisson recurrence times

1967 ◽  
Vol 4 (3) ◽  
pp. 605-608 ◽  
Author(s):  
Meyer Dwass
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variables ◽  
Recurrent Event ◽  
Recurrence Times ◽  
Image Position ◽  
Independent And Identically Distributed

Let Y1, Y2, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let Sn = Y1 + … + Yn, n = 1,2,…, S0 = 0. The event Sn = n is a recurrent event in the sense that successive waiting times between recurrences form a sequence of independent and identically distributed random variables. Specifically, the waiting time probabilities are (Alternately, the fn can be described as the probabilities for first return to the origin of the random walk whose successive steps are Y1 − 1, Y2 − 1, ….)

A limit theorem for random walks with drift

1967 ◽  
Vol 4 (1) ◽  
pp. 144-150 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
First Passage Time ◽  
Random Variables ◽  
Passage Time ◽  
Random Walk Process ◽  
First Passage ◽  
Time Out ◽  
Image Position

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

A limit theorem for random walks with drift

1967 ◽  
Vol 4 (01) ◽  
pp. 144-150 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
First Passage Time ◽  
Random Variables ◽  
Passage Time ◽  
Random Walk Process ◽  
First Passage ◽  
Time Out ◽  
Image Position

Let Xi , i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

scholarly journals Recurrence properties of Processes with stationary independent increments

1964 ◽  
Vol 4 (2) ◽  
pp. 223-228 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Increments ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

Let X1, X2,…Xn, … be independent and identically distributed random variables, and write . In [2] Chung and Fuchs have established necessary and sufficient conditions for the random walk {Zn} to be recurrent, i.e. for Zn to return infinitely often to every neighbourhood of the origin. The object of this paper is to obtain similar results for the corresponding process in continuous time.

A note on random walks in multidimensional time

1986 ◽  
Vol 99 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Janos Galambos ◽  
Imre Kátai
Keyword(s):  
Random Walks ◽  
Recent Work ◽  
Random Variables ◽  
Finite Variance ◽  
Image Position ◽  
Positive Integers ◽  
Independent And Identically Distributed

Let Kr denote the set of r-tuples n = (n1, n2, …, nr), r ≥ 1, where the components ni are positive integers. Let {X, Xn, n ∈ Kr} be a family of independent and identically distributed random variables with positive mean EX = μ < + ∞ and finite variance VX = σ2 < + ∞. In a recent work, M. Maejima and T. Mori [2] have shown that, if X is integer valued, aperiodic and E∣X∣3 < + ∞, then, for r = 2 or 3,wherethe summation being extended over all members j = (j1,j2, …, jr) of Kr that satisfy jt ≤ nt for all 1 ≤ t ≤ r.

A note on passage times and infinitely divisible distributions

1967 ◽  
Vol 4 (2) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Time Intervals ◽  
Mutually Independent ◽  
Passage Times

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

On a Class of Nonparametric Tests for Independence—Bivariate Case(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 337-342 ◽  
Author(s):  
M. S. Srivastava
Keyword(s):  
Random Variables ◽  
Nonparametric Tests ◽  
Absolutely Continuous ◽  
Bivariate Case ◽  
Image Position ◽  
Tests For Independence ◽  
Mutually Independent

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by(1)where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,(2)We wish to test the hypothesis(3)against the alternative(4)For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

Limiting diffusion for random walks with drift conditioned to stay positive

1978 ◽  
Vol 15 (02) ◽  
pp. 280-291 ◽  
Author(s):  
Peichuen Kao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Function ◽  
Random Variables ◽  
Principal Result ◽  
Hitting Time ◽  
Brownian Excursion ◽  
Norming Constant ◽  
Finite Dimensional ◽  
Independent Identically Distributed

Let {ξ k : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ 1} = μ ≠ 0. Form the random walk {S n : n ≧ 0} by setting S 0, S n = ξ 1 + ξ 2 + ··· + ξ n , n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ 1) that the finite-dimensional distributions of Xn , conditioned on n &lt; N &lt; ∞ converge to those of the Brownian excursion process.

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