scholarly journals Record statistics for a discrete-time random walk with correlated steps

2020 ◽  
Vol 2020 (2) ◽  
pp. 023206 ◽  
Author(s):  
Michael J Kearney
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Record Statistics

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states

1966 ◽  
Vol 3 (02) ◽  
pp. 403-434 ◽  
Author(s):  
E. Seneta ◽  
D. Vere-Jones
Keyword(s):  
Random Walk ◽  
Markov Chains ◽  
Recent Work ◽  
Discrete Time ◽  
Branching Process ◽  
Stationary Distributions

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.

scholarly journals Greedy Random Walk

2013 ◽  
Vol 23 (2) ◽  
pp. 269-289 ◽  
Author(s):  
TAL ORENSHTEIN ◽  
IGOR SHINKAR
Keyword(s):  
Random Walk ◽  
Random Process ◽  
Discrete Time ◽  
Complete Graph ◽  
Cover Time ◽  
Edge Cover ◽  
Adjacent Edge ◽  
Natural Families

We study a discrete time self-interacting random process on graphs, which we call greedy random walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not yet been crossed by the walker. At each step, the walker, being at some vertex, picks an adjacent edge among the edges that have not traversed thus far according to some (deterministic or randomized) rule. If all the adjacent edges have already been traversed, then an adjacent edge is chosen uniformly at random. After picking an edge the walker jumps along it to the neighbouring vertex. We show that the expected edge cover time of the greedy random walk is linear in the number of edges for certain natural families of graphs. Examples of such graphs include the complete graph, even degree expanders of logarithmic girth, and the hypercube graph. We also show that GRW is transient in$\mathbb{Z}^d$for alld≥ 3.

scholarly journals Stochastic stability of some state-dependent growth-collapse processes

2007 ◽  
Vol 39 (01) ◽  
pp. 189-220
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Power Law ◽  
Stochastic Stability ◽  
Long Range Dependence ◽  
Time Process ◽  
Power Law Distribution ◽  
State Dependent ◽  
Dependent Growth ◽  
Collapse Process

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

Amplitudes of Composition Fluctuations in the Simplest Chemical Reaction Equilibrium

2004 ◽  
Vol 218 (9) ◽  
pp. 1033-1040 ◽  
Author(s):  
M. Šolc ◽  
J. Hostomský
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Scale ◽  
Numerical Study ◽  
Mean Amplitude ◽  
Time Interval ◽  
The Mean ◽  
Composition Fluctuations ◽  
Number Of Particles

AbstractWe present a numerical study of equilibrium composition fluctuations in a system where the reaction X1 ⇔ X2 having the equilibrium constant equal to 1 takes place. The total number of reacting particles is N. On a discrete time scale, the amplitude of a fluctuation having the lifetime 2r reaction events is defined as the difference between the number of particles X1 in the microstate most distant from the microstate N/2 visited at least once during the fluctuation lifetime, and the equilibrium number of particles X1, N/2. On the discrete time scale, the mean value of this amplitude, m̅(r̅), is calculated in the random walk approximation. On a continuous time scale, the average amplitude of fluctuations chosen randomly and regardless of their lifetime from an ensemble of fluctuations occurring within the time interval (0,z), z → ∞, tends with increasing N to ~1.243 N0.25. Introducing a fraction of fluctuation lifetime during which the composition of the system spends below the mean amplitude m̅(r̅), we obtain a value of the mean amplitude of equilibrium fluctuations on the continuous time scale equal to ~1.19√N. The results suggest that using the random walk value m̅(r̅) and taking into account a) the exponential density of fluctuations lifetimes and b) the fact that the time sequence of reaction events represents the Poisson process, we obtain values of fluctuations amplitudes which differ only slightly from those derived for the Ehrenfest model.

scholarly journals Unified Solution of the Expected Maximum of a Discrete Time Random Walk and the Discrete Flux to a Spherical Trap

2006 ◽  
Vol 122 (5) ◽  
pp. 833-856 ◽  
Author(s):  
Satya N. Majumdar ◽  
Alain Comtet ◽  
Robert M. Ziff
Keyword(s):  
Random Walk ◽  
Discrete Time

Simple random walk statistics. Part I: Discrete time results

1996 ◽  
Vol 33 (2) ◽  
pp. 311-330 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Order Statistics ◽  
Discrete Time ◽  
Rank Order ◽  
Simple Random Walk ◽  
Alternative Method ◽  
Starting Point ◽  
Famous Paper ◽  
Rank Order Statistics

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.

scholarly journals Transient sequences in a hypernetwork generated by an adaptive network of spiking neurons

2017 ◽  
Vol 375 (2096) ◽  
pp. 20160288 ◽  
Author(s):  
Oleg V. Maslennikov ◽  
Dmitry S. Shchapin ◽  
Vladimir I. Nekorkin
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Spiking Neurons ◽  
Mathematical Methods ◽  
Adaptive Network ◽  
Upper Level ◽  
Simple Network ◽  
Dynamic States

We propose a model of an adaptive network of spiking neurons that gives rise to a hypernetwork of its dynamic states at the upper level of description. Left to itself, the network exhibits a sequence of transient clustering which relates to a traffic in the hypernetwork in the form of a random walk. Receiving inputs the system is able to generate reproducible sequences corresponding to stimulus-specific paths in the hypernetwork. We illustrate these basic notions by a simple network of discrete-time spiking neurons together with its FPGA realization and analyse their properties. This article is part of the themed issue ‘Mathematical methods in medicine: neuroscience, cardiology and pathology’.

Statevector reduction in discrete time: A random walk in Hilbert space

1994 ◽  
Vol 24 (3) ◽  
pp. 363-377 ◽  
Author(s):  
Akihiro Nakano ◽  
Philip Pearle
Keyword(s):  
Hilbert Space ◽  
Random Walk ◽  
Discrete Time
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