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Journal articles on the topic 'Random walk processses'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Dorogovtsev, A. A., and I. I. Nishchenko. "Loop-erased random walks associated with Markov processes." Theory of Stochastic Processes 25(41), no. 2 (December 11, 2021): 15–24. http://dx.doi.org/10.37863/tsp-1348277559-92.

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A new class of loop-erased random walks (LERW) on a finite set, defined as functionals from a Markov chain is presented. We propose a scheme in which, in contrast to the general settings of LERW, the loop-erasure is performed on a non-markovian sequence and moreover, not all loops are erased with necessity. We start with a special example of a random walk with loops, the number of which at every moment of time does not exceed a given fixed number. Further we consider loop-erased random walks, for which loops are erased at random moments of time that are hitting times for a Markov chain. The asymptotics of the normalized length of such loop-erased walks is established. We estimate also the speed of convergence of the normalized length of the loop-erased random walk on a finite group to the Rayleigh distribution.

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2

ABRAMSON, JOSH, and JIM PITMAN. "Concave Majorants of Random Walks and Related Poisson Processes." Combinatorics, Probability and Computing 20, no. 5 (August 18, 2011): 651–82. http://dx.doi.org/10.1017/s0963548311000307.

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We offer a unified approach to the theory of concave majorants of random walks, by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant of a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.

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3

Codling, Edward A., Michael J. Plank, and Simon Benhamou. "Random walk models in biology." Journal of The Royal Society Interface 5, no. 25 (April 15, 2008): 813–34. http://dx.doi.org/10.1098/rsif.2008.0014.

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Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

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4

Montero, Miquel. "Random Walks with Invariant Loop Probabilities: Stereographic Random Walks." Entropy 23, no. 6 (June 8, 2021): 729. http://dx.doi.org/10.3390/e23060729.

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Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

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5

Lindvall, Torgny, and L. C. G. Rogers. "On coupling of random walks and renewal processes." Journal of Applied Probability 33, no. 1 (March 1996): 122–26. http://dx.doi.org/10.2307/3215269.

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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.

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6

Lindvall, Torgny, and L. C. G. Rogers. "On coupling of random walks and renewal processes." Journal of Applied Probability 33, no. 01 (March 1996): 122–26. http://dx.doi.org/10.1017/s0021900200103778.

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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.

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7

YANG, ZHIHUI. "LARGE DEVIATION ASYMPTOTICS FOR RANDOM-WALK TYPE PERTURBATIONS." Stochastics and Dynamics 07, no. 01 (March 2007): 75–89. http://dx.doi.org/10.1142/s0219493707001950.

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Symmetric random walks can be arranged to converge to a Wiener process in the area of normal deviation. However, random walks and Wiener processes have, in general, different asymptotics of the large deviation probabilities. The action functionals for random-walks and Wiener processes are compared in this paper. The correction term is calculated. Exit problem and stochastic resonance for random-walk-type perturbation are also considered and compared with the white-noise-type perturbation.

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8

Nicolau, João. "STATIONARY PROCESSES THAT LOOK LIKE RANDOM WALKS— THE BOUNDED RANDOM WALK PROCESS IN DISCRETE AND CONTINUOUS TIME." Econometric Theory 18, no. 1 (February 2002): 99–118. http://dx.doi.org/10.1017/s0266466602181060.

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Several economic and financial time series are bounded by an upper and lower finite limit (e.g., interest rates). It is not possible to say that these time series are random walks because random walks are limitless with probability one (as time goes to infinity). Yet, some of these time series behave just like random walks. In this paper we propose a new approach that takes into account these ideas. We propose a discrete-time and a continuous-time process (diffusion process) that generate bounded random walks. These paths are almost indistinguishable from random walks, although they are stochastically bounded by an upper and lower finite limit. We derive for both cases the ergodic conditions, and for the diffusion process we present a closed expression for the stationary distribution. This approach suggests that many time series with random walk behavior can in fact be stationarity processes.

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9

Argyrakis, Panos. "Information dimension in random-walk processes." Physical Review Letters 59, no. 15 (October 12, 1987): 1729–32. http://dx.doi.org/10.1103/physrevlett.59.1729.

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10

Muminov, M., and E. G. Samandarov. "Inequalities for Some Random Walk Processes." Theory of Probability & Its Applications 30, no. 3 (September 1986): 489–95. http://dx.doi.org/10.1137/1130061.

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11

Karemera, David, and John Cole. "ARFIMA Tests for Random Walks in Exchange Rates in Asian, Latin American and African-Middle Eastern Markets." Review of Pacific Basin Financial Markets and Policies 13, no. 01 (March 2010): 1–18. http://dx.doi.org/10.1142/s0219091510001846.

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This article examines fractional processes as alternatives to random walks in emerging foreign exchange rate markets. Sowell's (1992) joint maximum likelihood is used to estimate the ARFIMA parameters and test for random walks. The results show that, in most cases, the emerging market exchange rates follow fractionally integrated processes. Forecasts of exchange rates based on the fractionally integrated autoregressive moving average models are compared to those from the benchmark random walk models. A Harvey, Leybourne and Newbold (1997) test of equality of forecast performance indicates that the ARFIMA forecasts are more efficient in the multi-step-ahead forecasts than the random walk model forecasts. The presence of fractional integration is seen to be associated with market inefficiency in the exchange markets examined. The evidence suggests that fractional integrated processes are viable alternatives to random walks for describing and forecasting exchange rates in the emerging markets.

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12

Georgiou, Nicholas, Mikhail V. Menshikov, Aleksandar Mijatović, and Andrew R. Wade. "Anomalous recurrence properties of many-dimensional zero-drift random walks." Advances in Applied Probability 48, A (July 2016): 99–118. http://dx.doi.org/10.1017/apr.2016.44.

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AbstractFamously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially nonhomogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension d≥2, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these elliptic random walks generalize the classical homogeneous Pearson‒Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.

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13

Guillotin-Plantard, Nadine, Françoise Pène, and Martin Wendler. "Empirical processes for recurrent and transient random walks in random scenery." ESAIM: Probability and Statistics 24 (2020): 127–37. http://dx.doi.org/10.1051/ps/2019030.

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In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.

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14

Neumann, Aneta, Bradley Alexander, and Frank Neumann. "Evolutionary Image Transition and Painting Using Random Walks." Evolutionary Computation 28, no. 4 (December 2020): 643–75. http://dx.doi.org/10.1162/evco_a_00270.

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We present a study demonstrating how random walk algorithms can be used for evolutionary image transition. We design different mutation operators based on uniform and biased random walks and study how their combination with a baseline mutation operator can lead to interesting image transition processes in terms of visual effects and artistic features. Using feature-based analysis we investigate the evolutionary image transition behaviour with respect to different features and evaluate the images constructed during the image transition process. Afterwards, we investigate how modifications of our biased random walk approaches can be used for evolutionary image painting. We introduce an evolutionary image painting approach whose underlying biased random walk can be controlled by a parameter influencing the bias of the random walk and thereby creating different artistic painting effects.

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15

Underwood, B. Y. "Random-walk modeling of turbulent impaction to a smooth wall." International Journal of Multiphase Flow 19, no. 3 (June 1993): 485–500. http://dx.doi.org/10.1016/0301-9322(93)90062-y.

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16

de Arcangelis, Lucilla, Antonio Coniglio, and Giovanni Paladin. "Comment on "Information Dimension in Random-Walk Processes"." Physical Review Letters 61, no. 18 (October 31, 1988): 2156. http://dx.doi.org/10.1103/physrevlett.61.2156.

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17

Doney, R. A., and E. M. Jones. "Conditioned random walks and Lévy processes." Bulletin of the London Mathematical Society 44, no. 1 (November 1, 2011): 139–50. http://dx.doi.org/10.1112/blms/bdr084.

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18

Berger, Noam, and Ron Rosenthal. "Random walks on discrete point processes." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 51, no. 2 (May 2015): 727–55. http://dx.doi.org/10.1214/13-aihp593.

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19

Abdullah, Mohammed, Colin Cooper, and Moez Draief. "Viral processes by random walks on random regular graphs." Annals of Applied Probability 25, no. 2 (April 2015): 477–522. http://dx.doi.org/10.1214/13-aap1000.

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20

Wasserman, Nicholas H. "A Random Walk: Stumbling across Connections." Mathematics Teacher 108, no. 9 (May 2015): 686–95. http://dx.doi.org/10.5951/mathteacher.108.9.0686.

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21

Takacs, Lajos. "Random Walk Processes and their Applications in Order Statistics." Annals of Applied Probability 2, no. 2 (May 1992): 435–59. http://dx.doi.org/10.1214/aoap/1177005710.

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22

Kim, Tae Yoon, and Sun Young Hwang. "Slow-explosive AR(1) processes converging to random walk." Communications in Statistics - Theory and Methods 49, no. 9 (February 16, 2019): 2094–109. http://dx.doi.org/10.1080/03610926.2019.1568486.

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23

Пиуновский, Алексей Борисович, and Alexei Borisovich Piunovskiy. "Turnpikes in finite Markov decision processes and random walk." Teoriya Veroyatnostei i ee Primeneniya 68, no. 1 (2023): 147–76. http://dx.doi.org/10.4213/tvp5528.

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В данной статье мы пересматриваем магистральное свойство для дисконтированных марковских процессов принятия решений, доказываем магистральную теорему для модели без дисконтирования и применяем полученные результаты для специфического случайного блуждания.

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24

Dong, Congzao, and Alexander Iksanov. "Weak convergence of random processes with immigration at random times." Journal of Applied Probability 57, no. 1 (March 2020): 250–65. http://dx.doi.org/10.1017/jpr.2019.88.

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AbstractBy a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

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25

Cammarota, Valentina, Valentina Cammarota, Энзо Орсингер, and Enzo Orsingher. "Angular processes related to Cauchy random walks." Teoriya Veroyatnostei i ee Primeneniya 55, no. 3 (2010): 489–506. http://dx.doi.org/10.4213/tvp4238.

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26

Cammarota, V., and E. Orsingher. "Angular Processes Related to Cauchy Random Walks." Theory of Probability & Its Applications 55, no. 3 (January 2011): 395–410. http://dx.doi.org/10.1137/s0040585x97984966.

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27

Albeverio, Sergio, and Frederik S. Herzberg. "Lifting Lévy processes to hyperfinite random walks." Bulletin des Sciences Mathématiques 130, no. 8 (December 2006): 697–706. http://dx.doi.org/10.1016/j.bulsci.2006.02.001.

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28

Louchard, G. "Random walks, Gaussian processes and list structures." Theoretical Computer Science 53, no. 1 (1987): 99–124. http://dx.doi.org/10.1016/0304-3975(87)90028-4.

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29

MAJID, S. "QUANTUM RANDOM WALKS AND TIME REVERSAL." International Journal of Modern Physics A 08, no. 25 (October 10, 1993): 4521–45. http://dx.doi.org/10.1142/s0217751x93001818.

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Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated with a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realized in Lin(H) in such a way that Δh=W(h⊗1)W−1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantum-mechanically to the system at time t+δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a novel kind of CTP theorem.

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30

Bertoin, Jean. "How linear reinforcement affects Donsker’s theorem for empirical processes." Probability Theory and Related Fields 178, no. 3-4 (September 18, 2020): 1173–92. http://dx.doi.org/10.1007/s00440-020-01001-9.

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Abstract A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

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31

Kalikova, A. "Statistical analysis of random walks on network." Scientific Journal of Astana IT University, no. 5 (July 27, 2021): 77–83. http://dx.doi.org/10.37943/aitu.2021.99.34.007.

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This paper describes an investigation of analytical formulas for parameters in random walks. Random walks are used to model situations in which an object moves in a sequence of steps in randomly chosen directions. Given a graph and a starting point, we select a neighbor of it at random, and move to this neighbor; then we select a neighbor of this point at random, and move to it etc. It is a fundamental dynamic process that arise in many models in mathematics, physics, informatics and can be used to model random processes inherent to many important applications. Different aspects of the theory of random walks on graphs are surveyed. In particular, estimates on the important parameters of hitting time, commute time, cover time are discussed in various works. In some papers, authors have derived an analytical expression for the distribution of the cover time for a random walk over an arbitrary graph that was tested for small values of n. However, this work will show the simplified analytical expressions for distribution of hitting time, commute time, cover time for bigger values of n. Moreover, this work will present the probability mass function and the cumulative distribution function for hitting time, commute time.

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32

Herrmann, Samuel, and Nicolas Massin. "Exit problem for Ornstein-Uhlenbeck processes: A random walk approach." Discrete & Continuous Dynamical Systems - B 25, no. 8 (2020): 3199–215. http://dx.doi.org/10.3934/dcdsb.2020058.

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33

Lototsky, Sergey, and Austin Pollok. "Kelly Criterion: From a Simple Random Walk to Lévy Processes." SIAM Journal on Financial Mathematics 12, no. 1 (January 2021): 342–68. http://dx.doi.org/10.1137/20m1330488.

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34

Mishkovski, Igor, Miroslav Mirchev, Sanja Scepanovic, and Ljupco Kocarev. "Interplay Between Spreading and Random Walk Processes in Multiplex Networks." IEEE Transactions on Circuits and Systems I: Regular Papers 64, no. 10 (October 2017): 2761–71. http://dx.doi.org/10.1109/tcsi.2017.2700948.

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35

Gorenflo, Rudolf, Gianni De Fabritiis, and Francesco Mainardi. "Discrete random walk models for symmetric Lévy–Feller diffusion processes." Physica A: Statistical Mechanics and its Applications 269, no. 1 (July 1999): 79–89. http://dx.doi.org/10.1016/s0378-4371(99)00082-5.

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36

Lee, Sang Bub, In Chan Kim, C. A. Miller, and S. Torquato. "Random-walk simulation of diffusion-controlled processes among static traps." Physical Review B 39, no. 16 (June 1, 1989): 11833–39. http://dx.doi.org/10.1103/physrevb.39.11833.

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37

Meerschaert, Mark M., and Peter Straka. "Semi-Markov approach to continuous time random walk limit processes." Annals of Probability 42, no. 4 (July 2014): 1699–723. http://dx.doi.org/10.1214/13-aop905.

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38

Guinard, Brieuc, and Amos Korman. "Intermittent inverse-square Lévy walks are optimal for finding targets of all sizes." Science Advances 7, no. 15 (April 2021): eabe8211. http://dx.doi.org/10.1126/sciadv.abe8211.

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Lévy walks are random walk processes whose step lengths follow a long-tailed power-law distribution. Because of their abundance as movement patterns of biological organisms, substantial theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantageous. However, despite extensive research, there is currently no mathematical proof indicating that Lévy walks are, in any manner, preferable strategies in higher dimensions than one. Here, we prove that in finite two-dimensional terrains, the inverse-square Lévy walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. Moreover, this holds even under the weak model of intermittent detection. Conversely, any other intermittent Lévy walk fails to efficiently find either large targets or small ones. Our results shed new light on the Lévy foraging hypothesis and are thus expected to affect future experiments on animals performing Lévy walks.

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39

Baek, Seung Ki, Hawoong Jeong, Seung-Woo Son, and Beom Jun Kim. "Zero-one-only process: A correlated random walk with a stochastic ratchet." International Journal of Modern Physics B 28, no. 29 (November 20, 2014): 1450201. http://dx.doi.org/10.1142/s0217979214502014.

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The investigation of random walks is central to a variety of stochastic processes in physics, chemistry and biology. To describe a transport phenomenon, we study a variant of the one-dimensional persistent random walk, which we call a zero-one-only process. It makes a step in the same direction as the previous step with probability p, and stops to change the direction with 1 − p. By using the generating-function method, we calculate its characteristic quantities such as the statistical moments and probability of the first return.

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40

Mazón, José M., Marcos Solera, and Julián Toledo. "Gradient flows in metric random walk spaces." SeMA Journal 79, no. 1 (October 10, 2021): 3–35. http://dx.doi.org/10.1007/s40324-021-00272-z.

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AbstractRecently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each $$x\in X$$ x ∈ X , which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.

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41

ISHIMURA, N., and N. YOSHIDA. "ON THE CONVERGENCE OF DISCRETE PROCESSES WITH MULTIPLE INDEPENDENT VARIABLES." ANZIAM Journal 58, no. 3-4 (March 6, 2017): 379–85. http://dx.doi.org/10.1017/s1446181116000389.

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We discuss discrete stochastic processes with two independent variables: one is the standard symmetric random walk, and the other is the Poisson process. Convergence of discrete stochastic processes is analysed, such that the symmetric random walk tends to the standard Brownian motion. We show that a discrete analogue of Ito’s formula converges to the corresponding continuous formula.

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42

Asmussen, Søren, and Sergey Foss. "On Exceedance Times for Some Processes with Dependent Increments." Journal of Applied Probability 51, no. 1 (March 2014): 136–51. http://dx.doi.org/10.1239/jap/1395771419.

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Let {Zn}n≥0 be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M = supn≥0Zn be its supremum. Asmussen and Klüppelberg (1996) considered the behavior of the random walk given that M > x for large x, and obtained a limit theorem, as x → ∞, for the distribution of the quadruple that includes the time τ = τ(x) to exceed level x, position Zτ at this time, position Zτ-1 at the prior time, and the trajectory up to it (similar results were obtained for the Cramér-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of τ. The class of models includes Markov-modulated models as particular cases. We also study fluid models, the Björk-Grandell risk process, give examples where the order of τ is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli and Schmidt (1999), Foss and Zachary (2002), and Foss, Konstantopoulos and Zachary (2007).

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43

Chen, Zhenlong, Lin Xu, and Dongjin Zhu. "Generalized continuous time random walks and Hermite processes." Statistics & Probability Letters 99 (April 2015): 44–53. http://dx.doi.org/10.1016/j.spl.2014.12.027.

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BERTOIN, J., and R. DONEY. "Spitzer's condition for random walks and Lévy Processes." Annales de l'Institut Henri Poincare (B) Probability and Statistics 33, no. 2 (1997): 167–78. http://dx.doi.org/10.1016/s0246-0203(97)80120-3.

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Borodin, Alexei, and Vadim Gorin. "Markov processes of infinitely many nonintersecting random walks." Probability Theory and Related Fields 155, no. 3-4 (March 14, 2012): 935–97. http://dx.doi.org/10.1007/s00440-012-0417-4.

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Chevyrev, Ilya. "Random walks and Lévy processes as rough paths." Probability Theory and Related Fields 170, no. 3-4 (May 17, 2017): 891–932. http://dx.doi.org/10.1007/s00440-017-0781-1.

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Abramson, Josh, Jim Pitman, Nathan Ross, and Geronimo Uribe Bravo. "Convex minorants of random walks and Lévy processes." Electronic Communications in Probability 16 (2011): 423–34. http://dx.doi.org/10.1214/ecp.v16-1648.

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Hara, Hiroaki, Ok Hee Chung, and Junji Koyama. "Dynamical activation processes described by generalized random walks." Physical Review B 46, no. 2 (July 1, 1992): 838–45. http://dx.doi.org/10.1103/physrevb.46.838.

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Vatutin, Vladimir A., and Elena E. Dyakonova. "Multitype weakly subcritical branching processes in random environment." Discrete Mathematics and Applications 31, no. 3 (June 1, 2021): 207–22. http://dx.doi.org/10.1515/dma-2021-0018.

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Abstract A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment X of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions E X < 0 and E XeX > 0.

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Brown, Mark, Erol A. Peköz, and Sheldon M. Ross. "SOME RESULTS FOR SKIP-FREE RANDOM WALK." Probability in the Engineering and Informational Sciences 24, no. 4 (August 19, 2010): 491–507. http://dx.doi.org/10.1017/s0269964810000136.

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A random walk that is skip-free to the left can only move down one level at a time but can skip up several levels. Such random walk features prominently in many places in applied probability including queuing theory and the theory of branching processes. This article exploits the special structure in this class of random walk to obtain a number of simplified derivations for results that are much more difficult in general cases. Although some of the results in this article have appeared elsewhere, our proof approach is different.

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