scholarly journals A renewal theorem and supremum of a perturbed random walk

2018 ◽  
Vol 23 ◽  
Author(s):  
Ewa Damek ◽  
Bartosz Kołodziejek
Keyword(s):  
Random Walk ◽  
Renewal Theorem ◽  
Perturbed Random Walk

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

scholarly journals PERFORMANCE MEASURES FOR THE TWO-NODE QUEUE WITH FINITE BUFFERS

2019 ◽  
Vol 34 (4) ◽  
pp. 522-549
Author(s):  
Yanting Chen ◽  
Xinwei Bai ◽  
Richard J. Boucherie ◽  
Jasper Goseling
Keyword(s):  
Random Walk ◽  
Performance Measures ◽  
Approximation Scheme ◽  
Queueing System ◽  
Original Model ◽  
Finite Capacity ◽  
Tandem Queue ◽  
Finite Buffers ◽  
Perturbed Random Walk ◽  
Markov Reward

We consider a two-node queue modeled as a two-dimensional random walk. In particular, we consider the case that one or both queues have finite buffers. We develop an approximation scheme based on the Markov reward approach to error bounds in order to bound performance measures of such random walks. The approximation scheme is developed in terms of a perturbed random walk in which the transitions along the boundaries are different from those in the original model and the invariant measure of the perturbed random walk is of product-form. We then apply this approximation scheme to a tandem queue and some variants of this model, for the case that both buffers are finite. The modified approximation scheme and the corresponding applications for a two-node queueing system in which only one of the buffers has finite capacity have also been discussed.

scholarly journals Tail asymptotics for the maximum of perturbed random walk

2006 ◽  
Vol 16 (3) ◽  
pp. 1411-1431 ◽  
Author(s):  
Victor F. Araman ◽  
Peter W. Glynn
Keyword(s):  
Random Walk ◽  
Tail Asymptotics ◽  
Perturbed Random Walk

Renewal Theorem for (L, 1)-Random Walk in Random Environment

2013 ◽  
Vol 33 (6) ◽  
pp. 1736-1748 ◽  
Author(s):  
Wenming HONG ◽  
Hongyan SUN
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Renewal Theorem

scholarly journals The renewal theorem for random walk in two-dimensional time

1972 ◽  
Vol 44 (1) ◽  
pp. 71-85 ◽  
Author(s):  
P. Ney ◽  
S. Wainger
Keyword(s):  
Random Walk ◽  
Two Dimensional ◽  
Renewal Theorem

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (01) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

scholarly journals Diffusion approximations for the maximum of a perturbed random walk

2005 ◽  
Vol 37 (3) ◽  
pp. 663-680
Author(s):  
Victor F. Araman ◽  
Peter W. Glynn
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Operations Management ◽  
Limit Theorems ◽  
Stationary Sequence ◽  
Risk Theory ◽  
Insurance Risk ◽  
Diffusion Approximations ◽  
Limit Theory ◽  
Perturbed Random Walk

Consider a random walk S=(Sn: n≥0) that is ‘perturbed’ by a stationary sequence (ξn: n≥0) to produce the process S=(Sn+ξn: n≥0). In this paper, we are concerned with developing limit theorems and approximations for the distribution of Mn=max{Sk+ξk: 0≤k≤n} when the random walk has a drift close to 0. Such maxima are of interest in several modeling contexts, including operations management and insurance risk theory. The associated limits combine features of both conventional diffusion approximations for random walks and extreme-value limit theory.

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