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.

A random walk analogue of Lévy’s Theorem

2008 ◽  
Vol 45 (2) ◽  
pp. 223-233 ◽  
Author(s):  
Takahiko Fujita
Keyword(s):  
Random Walk ◽  
Local Time ◽  
Stochastic Analysis ◽  
Type Theorem ◽  
Symmetric Random Walk ◽  
Absolute Value ◽  
Lévy’S Theorem ◽  
Definition Of ◽  
Skorokhod Reflection ◽  
Tanaka Formula

In this paper we will give a simple symmetric random walk analogue of Lévy’s Theorem. We will give a new definition of a local time of the simple symmetric random walk. We apply a discrete Itô formula to some absolute value like function to obtain a discrete Tanaka formula. Results in this paper rely upon a discrete Skorokhod reflection argument. This random walk analogue of Lévy’s theorem was already obtained by G. Simons ([14]) but it is still worth noting because we will use a discrete stochastic analysis to obtain it and this method is applicable to other research. We note some connection with previous results by Csáki, Révész, Csörgő and Szabados. Finally we observe that the discrete Lévy transformation in the present version is not ergodic. Lastly we give a Lévy-type theorem for simple nonsymmetric random walk using a discrete bang-bang process.

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

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (01) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

scholarly journals The Maximum of a Random Walk and Its Application to Rectangle Packing

1998 ◽  
Vol 12 (3) ◽  
pp. 373-386 ◽  
Author(s):  
E. G. Coffman ◽  
Philippe Flajolet ◽  
Leopold Flatto ◽  
Micha Hofri
Keyword(s):  
Random Walk ◽  
Upper Bound ◽  
Symmetric Random Walk ◽  
Rectangle Packing ◽  
Unit Width ◽  
Minimum Height ◽  
Image Position

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate,where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

Some joint distributions in Bernoulli excursions

1994 ◽  
Vol 31 (A) ◽  
pp. 239-250
Author(s):  
Endre Csáki
Keyword(s):  
Random Walk ◽  
Random Variables ◽  
Characteristic Functions ◽  
Symmetric Random Walk ◽  
Joint Distributions ◽  
Recursion Formulas

Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.

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