The correlated random walk with boundaries: A combinatorial solution

2000 ◽  
Vol 37 (2) ◽  
pp. 470-479 ◽  
Author(s):  
W. Böhm
Keyword(s):  
Random Walk ◽  
Asymptotic Formula ◽  
Lattice Paths ◽  
Correlated Random Walk ◽  
Absorbing Boundaries ◽  
Transition Functions ◽  
Special Cases ◽  
Combinatorial Construction

The transition functions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes.

The correlated random walk with boundaries: A combinatorial solution

2000 ◽  
Vol 37 (02) ◽  
pp. 470-479 ◽  
Author(s):  
W. Böhm
Keyword(s):  
Random Walk ◽  
Asymptotic Formula ◽  
Lattice Paths ◽  
Correlated Random Walk ◽  
Absorbing Boundaries ◽  
Transition Functions ◽  
Special Cases ◽  
Combinatorial Construction

The transition functions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes.

Correlated random walk

1955 ◽  
Vol 51 (4) ◽  
pp. 639-651 ◽  
Author(s):  
J. Gillis
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Correlated Random Walk ◽  
Dimensional Lattice ◽  
General Properties ◽  
Special Cases ◽  
Previous Step

ABSTRACTRandom walk on a d-dimensional lattice is investigated such that, at any stage, the probabilities of the step being in the various possible directions depend upon the direction of the previous step. The motion may be characterized by a generating function which is here derived. The generating function is then used to obtain some general properties of the walk. Certain special cases are considered in greater detail. The existence of recurrent points is investigated in particular, and the probability of returning to the origin after 2n steps. This latter function is evaluated asymptotically for the cases d = 1 and d = an even integer.

A compass without a map: tortuosity and orientation of eastern painted turtles (Chrysemys picta picta) released in unfamiliar territory

2006 ◽  
Vol 84 (8) ◽  
pp. 1129-1137 ◽  
Author(s):  
I.R. Caldwell ◽  
V.O. Nams
Keyword(s):  
Fractal Dimension ◽  
Random Walk ◽  
Spatial Scales ◽  
Chrysemys Picta ◽  
Reference Stimulus ◽  
Correlated Random Walk ◽  
Specific Mechanism ◽  
Minimal Time ◽  
Painted Turtles

Orientation mechanisms allow animals to spend minimal time in hostile areas while reaching needed resources. Identification of the specific mechanism used by an animal can be difficult, but examining an animal's path in familiar and unfamiliar areas can provide clues to the type of mechanism in use. Semiaquatic turtles are known to use a homing mechanism in familiar territory to locate their home lake while on land, but little is known about their ability to locate habitat in unfamiliar territory. We tested the tortuosity and orientation of 60 eastern painted turtles ( Chrysemys picta picta (Schneider, 1783)). We released turtles at 20 release points located at five distances and in two directions from two unfamiliar lakes. Turtle trails were quite straight (fractal dimension between 1.1 and 1.025) but were not oriented towards water from any distance (V-test; u < 0.72; P > 0.1). Turtles maintained their initially chosen direction but either could not detect water or were not motivated to reach it. Furthermore, paths were straighter at larger spatial scales than at smaller spatial scales, which could not have occurred if the turtles had been using a correlated random walk. Turtles must therefore be using a reference stimulus for navigation even in unfamiliar areas.

Optimal stopping rules for correlated random walks with a discount

2004 ◽  
Vol 41 (2) ◽  
pp. 483-496 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Optimal Stopping ◽  
Constant Factor ◽  
Stopping Rules ◽  
Correlated Random Walk ◽  
Numerical Examples ◽  
Optimal Rule ◽  
Correlated Random Walks ◽  
Optimal Stopping Rules

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

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.

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.

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