A ratio limit theorem for symmetric random walk

1970 ◽  
Vol 23 (1) ◽  
pp. 199-213 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Symmetric Random Walk ◽  
Ratio Limit

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

A Berry-Esseen Type Theorem on Nilpotent Covering Graphs

2004 ◽  
Vol 56 (5) ◽  
pp. 963-982 ◽  
Author(s):  
Satoshi Ishiwata
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Type Theorem ◽  
Speed Of Convergence ◽  
Complete Proof ◽  
Symmetric Random Walk ◽  
Markov Kernel ◽  
Covering Graph

AbstractWe prove an estimate for the speed of convergence of the transition probability for a symmetric random walk on a nilpotent covering graph. To obtain this estimate, we give a complete proof of the Gaussian bound for the gradient of the Markov kernel.

Some Results in a Correlated Random Walk

1971 ◽  
Vol 14 (3) ◽  
pp. 341-347 ◽  
Author(s):  
G. C. Jain
Keyword(s):  
Random Walk ◽  
Joint Distribution ◽  
Statistical Problem ◽  
Correlated Random Walk ◽  
Symmetric Random Walk

In connection with a statistical problem concerning the Galtontest Cśaki and Vincze [1] gave for an equivalent Bernoullian symmetric random walk the joint distribution of g and k, denoting respectively the number of positive steps and the number of times the particle crosses the origin, given that it returns there on the last step.

scholarly journals Two explicit Skorokhod embeddings for simple symmetric random walk

2019 ◽  
Vol 129 (9) ◽  
pp. 3431-3445 ◽  
Author(s):  
Xue Dong He ◽  
Sang Hu ◽  
Jan Obłój ◽  
Xun Yu Zhou
Keyword(s):  
Random Walk ◽  
Symmetric Random Walk
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