scholarly journals Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk

2019 ◽  
Vol 33 (2) ◽  
pp. 356-373
Author(s):  
Bastien Mallein
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Branching Random Walk ◽  
Maximal Displacement ◽  
Necessary And Sufficient

Overshoots over curved boundaries

2003 ◽  
Vol 35 (2) ◽  
pp. 417-448 ◽  
Author(s):  
R. A. Doney ◽  
P. S. Griffin
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Curved Boundaries ◽  
Symmetric Region ◽  
Necessary And Sufficient

We consider the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n) :|x| ≤ rnb} as r → ∞. In order to be sure that this actually occurs, we treat only the case where the power b lies in the interval [0,½), and we further assume a condition that prevents the probability of exiting at either boundary tending to 0. Under these restrictions we establish necessary and sufficient conditions for the pth moment of the overshoot to be O(rq), and for the overshoot to be tight, as r → ∞.

On the rate of growth of the overshoot and the maximum partial sum

1998 ◽  
Vol 30 (1) ◽  
pp. 181-196 ◽  
Author(s):  
P. S. Griffin ◽  
R. A. Maller
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Strong Sense ◽  
Partial Sum ◽  
Order Of Magnitude ◽  
The Stability ◽  
Dominance Properties ◽  
Necessary And Sufficient ◽  
First Time

Let Tr be the first time at which a random walk Sn escapes from the strip [-r,r], and let |STr|-r be the overshoot of the boundary of the strip. We investigate the order of magnitude of the overshoot, as r → ∞, by providing necessary and sufficient conditions for the ‘stability’ of |STr|, by which we mean that |STr|/r converges to 1, either in probability (weakly) or almost surely (strongly), as r → ∞. These also turn out to be equivalent to requiring only the boundedness of |STr|/r, rather than its convergence to 1, either in the weak or strong sense, as r → ∞. The almost sure characterisation turns out to be extremely simple to state and to apply: we have |STr|/r → 1 a.s. if and only if EX2 < ∞ and EX = 0 or 0 < |EX| ≤ E|X| < ∞. Proving this requires establishing the equivalence of the stability of STr with certain dominance properties of the maximum partial sum Sn* = max{|Sj|: 1 ≤ j ≤ n} over its maximal increment.

On the rate of growth of the overshoot and the maximum partial sum

1998 ◽  
Vol 30 (01) ◽  
pp. 181-196 ◽  
Author(s):  
P. S. Griffin ◽  
R. A. Maller
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Strong Sense ◽  
Partial Sum ◽  
Order Of Magnitude ◽  
The Stability ◽  
Dominance Properties ◽  
Necessary And Sufficient ◽  
First Time

Let T r be the first time at which a random walk S n escapes from the strip [-r,r], and let |S T r |-r be the overshoot of the boundary of the strip. We investigate the order of magnitude of the overshoot, as r → ∞, by providing necessary and sufficient conditions for the ‘stability’ of |S T r |, by which we mean that |S T r |/r converges to 1, either in probability (weakly) or almost surely (strongly), as r → ∞. These also turn out to be equivalent to requiring only the boundedness of |S T r |/r, rather than its convergence to 1, either in the weak or strong sense, as r → ∞. The almost sure characterisation turns out to be extremely simple to state and to apply: we have |S T r |/r → 1 a.s. if and only if EX 2 &lt; ∞ and EX = 0 or 0 &lt; |EX| ≤ E|X| &lt; ∞. Proving this requires establishing the equivalence of the stability of S T r with certain dominance properties of the maximum partial sum S n * = max{|S j |: 1 ≤ j ≤ n} over its maximal increment.

scholarly journals STRUCTURE-REVERSIBILITY OF A TWO-DIMENSIONAL REFLECTING RANDOM WALK AND ITS APPLICATION TO QUEUEING NETWORK

2014 ◽  
Vol 29 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Masahiro Kobayashi ◽  
Masakiyo Miyazawa ◽  
Hiroshi Shimizu
Keyword(s):  
Random Walk ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Queueing Network ◽  
Necessary And Sufficient Conditions ◽  
Negative Integer ◽  
Product Form ◽  
Two Dimensional ◽  
Necessary And Sufficient

We consider a two-dimensional reflecting random walk on the non-negative integer quadrant. It is assumed that this reflecting random walk has skip-free transitions. We are concerned with its time-reversed process assuming that the stationary distribution exists. In general, the time-reversed process may not be a reflecting random walk. In this paper, we derive necessary and sufficient conditions for the time-reversed process also to be a reflecting random walk. These conditions are different from but closely related to the product form of the stationary distribution.

Overshoots over curved boundaries. II

2004 ◽  
Vol 36 (4) ◽  
pp. 1148-1174 ◽  
Author(s):  
R. A. Doney ◽  
P. S. Griffin
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Curved Boundaries ◽  
Symmetric Region ◽  
Necessary And Sufficient ◽  
Upper Boundary

We continue the study of the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n): |x| ≤ rnb} as r → ∞ which was begun in Part I of this work. In contrast to that paper, we are interested in the case where the probability of exiting at the upper boundary tends to 1. In this scenario we treat the case where the power b lies in the interval [0, 1), and we establish necessary and sufficient conditions for the overshoot to be relatively stable in probability (except for the case ), and for the pth moment of the overshoot to be O(rq) as r → ∞.

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.

Overshoots over curved boundaries

2003 ◽  
Vol 35 (02) ◽  
pp. 417-448 ◽  
Author(s):  
R. A. Doney ◽  
P. S. Griffin
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Curved Boundaries ◽  
Symmetric Region ◽  
Necessary And Sufficient

We consider the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n) :|x| ≤ rn b } as r → ∞. In order to be sure that this actually occurs, we treat only the case where the power b lies in the interval [0,½), and we further assume a condition that prevents the probability of exiting at either boundary tending to 0. Under these restrictions we establish necessary and sufficient conditions for the pth moment of the overshoot to be O(r q ), and for the overshoot to be tight, as r → ∞.

Overshoots over curved boundaries. II

2004 ◽  
Vol 36 (04) ◽  
pp. 1148-1174
Author(s):  
R. A. Doney ◽  
P. S. Griffin
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Curved Boundaries ◽  
Symmetric Region ◽  
Necessary And Sufficient ◽  
Upper Boundary

We continue the study of the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n): |x| ≤ rn b } as r → ∞ which was begun in Part I of this work. In contrast to that paper, we are interested in the case where the probability of exiting at the upper boundary tends to 1. In this scenario we treat the case where the power b lies in the interval [0, 1), and we establish necessary and sufficient conditions for the overshoot to be relatively stable in probability (except for the case ), and for the pth moment of the overshoot to be O(r q ) as r → ∞.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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