scholarly journals On a class of random walks in simplexes

2020 ◽  
Vol 57 (2) ◽  
pp. 409-428
Author(s):  
Tuan-Minh Nguyen ◽  
Stanislav Volkov
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Beta Distribution ◽  
Interior Point ◽  
Random Variable ◽  
Limiting Distributions ◽  
Special Cases ◽  
Limit Behaviour ◽  
New Location

AbstractWe study the limit behaviour of a class of random walk models taking values in the standard d-dimensional ( $d\ge 1$ ) simplex. From an interior point z, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on z, and then the particle randomly jumps to a new location z′ on the segment connecting z to the chosen vertex. In some special cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are Dirichlet. We also consider a related history-dependent random walk model in [0, 1] based on an urn-type scheme. We show that this random walk converges in distribution to an arcsine random variable.

On the number of jumps of random walks with a barrier

2008 ◽  
Vol 40 (01) ◽  
pp. 206-228 ◽  
Author(s):  
Alex Iksanov ◽  
Martin Möhle
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Random Variable ◽  
Tail Behavior ◽  
Limiting Distributions ◽  
Current State ◽  
Single Block ◽  
The Right ◽  
First Time

LetS0:= 0 andSk:=ξ1+ ··· +ξkfork∈ ℕ := {1, 2, …}, where {ξk:k∈ ℕ} are independent copies of a random variableξwith values in ℕ and distributionpk:= P{ξ=k},k∈ ℕ. We interpret the random walk {Sk:k= 0, 1, 2, …} as a particle jumping to the right through integer positions. Fixn∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal ton. This constraint defines an increasing Markov chain {Rk(n):k= 0, 1, 2, …} which never reaches the staten. We call this process a random walk with barriern. LetMndenote the number of jumps of the random walk with barriern. This paper focuses on the asymptotics ofMnasntends to ∞. A key observation is that, underp1> 0, {Mn:n∈ ℕ} satisfies the distributional recursionM1= 0 andforn= 2, 3, …, whereInis independent ofM2, …,Mn−1with distribution P{In=k} =pk/ (p1+ ··· +pn−1),k∈ {1, …,n− 1}. Depending on the tail behavior of the distribution ofξ, several scalings forMnand corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the number of jumps,Mn, with the first time,Nn, when the unrestricted random walk {Sk:k= 0, 1, …} reaches a state larger than or equal ton. The results are applied to derive the asymptotics of the number of collision events (that take place until there is just a single block) forβ(a,b)-coalescent processes with parameters 0 <a< 2 andb= 1.

scholarly journals On the number of jumps of random walks with a barrier

2008 ◽  
Vol 40 (1) ◽  
pp. 206-228 ◽  
Author(s):  
Alex Iksanov ◽  
Martin Möhle
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Random Variable ◽  
Tail Behavior ◽  
Limiting Distributions ◽  
Current State ◽  
Single Block ◽  
The Right ◽  
First Time

Let S0 := 0 and Sk := ξ1 + ··· + ξk for k ∈ ℕ := {1, 2, …}, where {ξk : k ∈ ℕ} are independent copies of a random variable ξ with values in ℕ and distribution pk := P{ξ = k}, k ∈ ℕ. We interpret the random walk {Sk : k = 0, 1, 2, …} as a particle jumping to the right through integer positions. Fix n ∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal to n. This constraint defines an increasing Markov chain {Rk(n) : k = 0, 1, 2, …} which never reaches the state n. We call this process a random walk with barrier n. Let Mn denote the number of jumps of the random walk with barrier n. This paper focuses on the asymptotics of Mn as n tends to ∞. A key observation is that, under p1 > 0, {Mn : n ∈ ℕ} satisfies the distributional recursion M1 = 0 and for n = 2, 3, …, where In is independent of M2, …, Mn−1 with distribution P{In = k} = pk / (p1 + ··· + pn−1), k ∈ {1, …, n − 1}. Depending on the tail behavior of the distribution of ξ, several scalings for Mn and corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the number of jumps, Mn, with the first time, Nn, when the unrestricted random walk {Sk : k = 0, 1, …} reaches a state larger than or equal to n. The results are applied to derive the asymptotics of the number of collision events (that take place until there is just a single block) for β(a, b)-coalescent processes with parameters 0 < a < 2 and b = 1.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
Author(s):  
M. Jara ◽  
T. Komorowski
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Transport ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transport Theory ◽  
Sufficient Conditions ◽  
Continuous Time Random Walks ◽  
Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

scholarly journals One dimensional quantum walks with memory

2010 ◽  
Vol 10 (5&6) ◽  
pp. 509-524
Author(s):  
M. Mc Gettrick
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Walks ◽  
One Dimensional ◽  
With Memory ◽  
Previous Step

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of one previous step. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

Series expansions for the all-time maximum of α-stable random walks

2016 ◽  
Vol 48 (3) ◽  
pp. 744-767
Author(s):  
Clifford Hurvich ◽  
Josh Reed
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Queueing Theory ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Series Expansions ◽  
Stable Random Variables ◽  
The Mean ◽  
The Stability

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

Determining Equivalent Resistance and Nodal Potentials of a Resistive Circuit Using a Reduced Transition Matrix

2020 ◽  
Vol 02 (01) ◽  
pp. 2050004
Author(s):  
Je-Young Choi
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Transition Matrix ◽  
Present Approach ◽  
Voltage Source ◽  
Equivalent Resistance ◽  
Electrical Circuits ◽  
Electric Potentials ◽  
Dominant Eigenvector

Several methods have been developed in order to solve electrical circuits consisting of resistors and an ideal voltage source. A correspondence with random walks avoids difficulties caused by choosing directions of currents and signs in potential differences. Starting from the random-walk method, we introduce a reduced transition matrix of the associated Markov chain whose dominant eigenvector alone determines the electric potentials at all nodes of the circuit and the equivalent resistance between the nodes connected to the terminals of the voltage source. Various means to find the eigenvector are developed from its definition. A few example circuits are solved in order to show the usefulness of the present approach.

The existence of moments for stationary Markov chains

1983 ◽  
Vol 20 (01) ◽  
pp. 191-196 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Time Dependent ◽  
General Function ◽  
Convergence Of Moments ◽  
Special Cases

We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr and f(x) = esx . In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

scholarly journals The connection between evolution algebras, random walks and graphs

2019 ◽  
Vol 19 (02) ◽  
pp. 2050023 ◽  
Author(s):  
Paula Cadavid ◽  
Mary Luz Rodiño Montoya ◽  
Pablo M. Rodriguez
Keyword(s):  
Probability Theory ◽  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Associative Algebras ◽  
Biological Phenomena ◽  
Evolution Algebra ◽  
New Type ◽  
Markov Evolution

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras.

scholarly journals On the Moments of Hitting Times for Random Walks on Trees

2009 ◽  
Vol 2009 ◽  
pp. 1-4 ◽  
Author(s):  
José Luis Palacios
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Hitting Times ◽  
Natural Numbers ◽  
Ergodic Markov Chain

Using classical arguments we derive a formula for the moments of hitting times for an ergodic Markov chain. We apply this formula to the case of simple random walk on trees and show, with an elementary electric argument, that all the moments are natural numbers.

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