Random walks on discrete and continuous circles

1993 ◽  
Vol 30 (4) ◽  
pp. 780-789 ◽  
Author(s):  
Jeffrey S. Rosenthal
Keyword(s):  
Random Walk ◽  
Fourier Analysis ◽  
Random Walks ◽  
Large Class ◽  
Lipschitz Function ◽  
Generation Gap

We consider a large class of random walks on the discrete circle Z/(n), defined in terms of a piecewise Lipschitz function, and motivated by the ‘generation gap' process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently of n. Our techniques involve Fourier analysis and a comparison of the random walks on Z/(n) with a random walk on the continuous circle S1.

Random walks on discrete and continuous circles

1993 ◽  
Vol 30 (04) ◽  
pp. 780-789
Author(s):  
Jeffrey S. Rosenthal
Keyword(s):  
Random Walk ◽  
Fourier Analysis ◽  
Random Walks ◽  
Large Class ◽  
Lipschitz Function ◽  
Generation Gap

We consider a large class of random walks on the discrete circle Z/(n), defined in terms of a piecewise Lipschitz function, and motivated by the ‘generation gap' process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently of n. Our techniques involve Fourier analysis and a comparison of the random walks on Z/(n) with a random walk on the continuous circle S 1.

scholarly journals Likelihood Orders for the $p$-Cycle Walks on the Symmetric Group

10.37236/7373 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Megan Bernstein
Keyword(s):  
Random Walk ◽  
Fourier Analysis ◽  
Symmetric Group ◽  
Random Walks ◽  
Total Variation ◽  
Separation Distance ◽  
Mixing Times ◽  
Sufficient Time ◽  
Discrete Fourier Analysis ◽  
P Cycle

Consider for a random walk on a group, the order from most to least likely element of the walk at each step, called the likelihood order. Up to periodicity issues, this order stabilizes after a sufficient number of steps. Here discrete Fourier analysis and the representations of the symmetric group, particularly formulas for the characters, are used to find the order after sufficient time for the random walks on the symmetric group generated by $p$-cycles for any $p$ fixed, $n$ sufficiently large. For the transposition walk, generated by all the $2$-cycles, at various levels of laziness, it is shown that order $n^2$ steps suffice for the order to stabilize.  Likelihood orders can aid in finding the total variation or separation distance mixing times.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (2) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

On the number of times where a simple random walk reaches its maximum

1992 ◽  
Vol 29 (02) ◽  
pp. 305-312 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Probability And Statistics

Let Qn denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn , are considered and their asymptotic behavior is studied.

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 Random-Walk, Agent-Level Pandemic Simulation (RAW-ALPS) for Analyzing Effects of Different Lockdown Measures

2021 ◽  
Vol 7 ◽  
Author(s):  
Anuj Srivastava
Keyword(s):  
Random Walk ◽  
Simulation Model ◽  
Random Walks ◽  
Stochastic Simulation ◽  
Preventive Measures ◽  
Infection Rates ◽  
Physical Proximity ◽  
Stochastic Simulation Model ◽  
Restrictive Measures ◽  
Effective Use

This article develops an agent-level stochastic simulation model, termed RAW-ALPS, for simulating the spread of an epidemic in a community. The mechanism of transmission is agent-to-agent contact, using parameters reported for the COVID-19 pandemic. When unconstrained, the agents follow independent random walks and catch infections due to physical proximity with infected agents. Under lockdown, an infected agent can only infect a coinhabitant, leading to a reduction in the spread. The main goal of the RAW-ALPS simulation is to help quantify the effects of preventive measures—timing and durations of lockdowns—on infections, fatalities, and recoveries. The model helps measure changes in infection rates and casualties due to the imposition and maintenance of restrictive measures. It considers three types of lockdowns: 1) whole population (except the essential workers), 2) only the infected agents, and 3) only the symptomatic agents. The results show that the most effective use of lockdown measures is when all infected agents, including both symptomatic and asymptomatic, are quarantined, while the uninfected agents are allowed to move freely. This result calls for regular and extensive testing of a population to isolate and restrict all infected agents.

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