The general correlated random walk

1994 ◽  
Vol 31 (04) ◽  
pp. 869-884 ◽  
Author(s):  
Anyue Chen ◽  
Eric Renshaw
Keyword(s):  
Random Walk ◽  
Characteristic Function ◽  
Recurrence Formula ◽  
Correlated Random Walk

In this paper we provide further results on the generald-dimensional correlated random walk. In particular, we prove that then-step characteristic function of any correlated random walk satisfies a recurrence formula which enables both it and the total characteristic function to be obtained. Some examples are then considered.

The general correlated random walk

1994 ◽  
Vol 31 (4) ◽  
pp. 869-884 ◽  
Author(s):  
Anyue Chen ◽  
Eric Renshaw
Keyword(s):  
Random Walk ◽  
Characteristic Function ◽  
Recurrence Formula ◽  
Correlated Random Walk

In this paper we provide further results on the general d-dimensional correlated random walk. In particular, we prove that the n-step characteristic function of any correlated random walk satisfies a recurrence formula which enables both it and the total characteristic function to be obtained. Some examples are then considered.

The Gillis–Domb–Fisher correlated random walk

1992 ◽  
Vol 29 (04) ◽  
pp. 792-813 ◽  
Author(s):  
Anyue Chen ◽  
Eric Renshaw
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Characteristic Function ◽  
Simple Proof ◽  
Complete Information ◽  
Initial Distribution ◽  
Fundamental Importance ◽  
Correlated Random Walk ◽  
Scientific Disciplines ◽  
Probability Structure

Correlated random walk models figure prominently in many scientific disciplines. Of fundamental importance in such applications is the development of the characteristic function of then-step probability distribution since it contains complete information on the probability structure of the process. Using a simple algebraic lemma we derive then-step characteristic function of the Gillis correlated random walk together with other related results. In particular, we present a new and simple proof of Gillis's conjecture, consider the generalization to the Gillis–Domb–Fisher walk, and examine the effect of including an arbitrary initial distribution.

The Gillis–Domb–Fisher correlated random walk

1992 ◽  
Vol 29 (4) ◽  
pp. 792-813 ◽  
Author(s):  
Anyue Chen ◽  
Eric Renshaw
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Characteristic Function ◽  
Simple Proof ◽  
Complete Information ◽  
Initial Distribution ◽  
Fundamental Importance ◽  
Correlated Random Walk ◽  
Scientific Disciplines ◽  
Probability Structure

Correlated random walk models figure prominently in many scientific disciplines. Of fundamental importance in such applications is the development of the characteristic function of the n-step probability distribution since it contains complete information on the probability structure of the process. Using a simple algebraic lemma we derive the n-step characteristic function of the Gillis correlated random walk together with other related results. In particular, we present a new and simple proof of Gillis's conjecture, consider the generalization to the Gillis–Domb–Fisher walk, and examine the effect of including an arbitrary initial distribution.

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.

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