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@article{pzSL1, author = {Peterson, Jonathon and Zeitouni, Ofer}, coden = {APBYAE}, doi = {10.1214/08-AOP399}, eprint = {0704.1778}, fjournal = {The Annals of Probability}, issn = {0091-1798}, journal = {Ann. Probab.}, mrclass = {60K37 (60F05 82C41 92D30)}, mrnumber = {2489162 (2010g:60222)}, mrreviewer = {Firas Rassoul-Agha}, number = {1}, pages = {143--188}, title = {{Quenched limits for transient, zero speed one-dimensional random walk in random environment}}, url = {http://dx.doi.org/10.1214/08-AOP399}, volume = {37}, year = {2009} }

@article{p1LSL2, author = {Peterson, Jonathon}, doi = {10.1214/08-AIHP149}, eprint = {0708.0649}, fjournal = {Annales de l'Institut Henri Poincar{\'e} Probabilit{\'e}s et Statistiques}, issn = {0246-0203}, journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.}, mrclass = {60K37 (60F05 82C41)}, mrnumber = {2548499 (2011f:60206)}, mrreviewer = {Cl{\'e}ment Dombry}, number = {3}, pages = {685--709}, title = {{Quenched limits for transient, ballistic, sub-{G}aussian one-dimensional random walk in random environment}}, url = {http://dx.doi.org/10.1214/08-AIHP149}, volume = {45}, year = {2009} }

@article{pzLDPRWRE, author = {Peterson, Jonathon and Zeitouni, Ofer}, eprint = {0812.3619}, fjournal = {ALEA. Latin American Journal of Probability and Mathematical Statistics}, issn = {1980-0436}, journal = {ALEA Lat. Am. J. Probab. Math. Stat.}, mrclass = {60F10 (60G50 60K37)}, mrnumber = {2557875 (2011b:60100)}, mrreviewer = {Dimitris Cheliotis}, pages = {349--368}, title = {{On the annealed large deviation rate function for a multi-dimensional random walk in random environment}}, url = {http://alea.impa.br/articles/v6/06-15.pdf}, volume = {6}, year = {2009} }

@article{pRWRESystem, author = {Peterson, Jonathon}, doi = {10.1214/EJP.v15-784}, eprint = {0907.3680}, fjournal = {Electronic Journal of Probability}, issn = {1083-6489}, journal = {Electron. J. Probab.}, mrclass = {60K37 (60F10 60K35)}, mrnumber = {2659756 (2011f:60207)}, mrreviewer = {Jean-Baptiste Bardet}, pages = {no. 32, 1024--1040}, title = {{Systems of one-dimensional random walks in a common random environment}}, url = {http://dx.doi.org/10.1214/EJP.v15-784}, volume = {15}, year = {2010} }

@article{psRWRECurrent, author = {Peterson, Jonathon and Sepp{\"a}l{\"a}inen, Timo}, coden = {APBYAE}, doi = {10.1214/10-AOP537}, eprint = {0904.4768}, fjournal = {The Annals of Probability}, issn = {0091-1798}, journal = {Ann. Probab.}, mrclass = {60K37 (60K35)}, mrnumber = {2683630 (2011k:60334)}, mrreviewer = {Marcel Ortgiese}, number = {6}, pages = {2258--2294}, title = {{Current fluctuations of a system of one-dimensional random walks in random environment}}, url = {http://dx.doi.org/10.1214/10-AOP537}, volume = {38}, year = {2010} }

@article{pCPRE, author = {Peterson, Jonathon}, coden = {STOPB7}, doi = {10.1016/j.spa.2010.11.003}, eprint = {1005.0810}, fjournal = {Stochastic Processes and their Applications}, issn = {0304-4149}, journal = {Stochastic Process. Appl.}, mrclass = {60K35 (05C80 60K37)}, mrnumber = {2763098 (2012c:60238)}, mrreviewer = {Nicolas Lanchier}, number = {3}, pages = {609--629}, title = {{The contact process on the complete graph with random vertex-dependent infection rates}}, url = {http://dx.doi.org/10.1016/j.spa.2010.11.003}, volume = {121}, year = {2011} }

@article{gpRWREBridges, author = {Gantert, Nina and Peterson, Jonathon}, doi = {10.1214/10-AIHP378}, eprint = {0910.4927}, fjournal = {Annales de l'Institut Henri Poincar{\'e} Probabilit{\'e}s et Statistiques}, issn = {0246-0203}, journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.}, mrclass = {60K37}, mrnumber = {2841070 (2012h:60311)}, mrreviewer = {Bernardo D'Auria}, number = {3}, pages = {663--678}, title = {{Maximal displacement for bridges of random walks in a random environment}}, url = {http://dx.doi.org/10.1214/10-AIHP378}, volume = {47}, year = {2011} }

@article{psWQLXn, author = {Peterson, Jonathon and Samorodnitsky, Gennady}, eprint = {1112.3919}, fjournal = {ALEA. Latin American Journal of Probability and Mathematical Statistics}, journal = {ALEA Lat. Am. J. Probab. Math. Stat.}, number = {2}, pages = {531--569}, title = {{Weak weak quenched limits for the path-valued processes of hitting times and positions of a transient, one-dimensional random walk in a random environment}}, url = {http://alea.impa.br/articles/v9/09-22.pdf}, volume = {9}, year = {2012} }

@article{pLDPERW, author = {Peterson, Jonathon}, doi = {10.1214/EJP.v17-1726}, eprint = {1201.0318}, fjournal = {Electronic Journal of Probability}, issn = {1083-6489}, journal = {Electron. J. Probab.}, mrclass = {60F10 (60K37)}, mrnumber = {2946155}, mrreviewer = {Ofer Zeitouni}, number = {48}, pages = {1--24}, title = {{Large deviations and slowdown asymptotics for one-dimensional excited random walks}}, url = {http://dx.doi.org/10.1214/EJP.v17-1726}, volume = {17}, year = {2012} }

@article{pCRWMono, author = {Peterson, Jonathon}, eprint = {1210.4518}, fjournal = {Markov Processes and Related Fields}, journal = {Markov Process. Related Fields}, number = {4}, pages = {721--734}, title = {{Strict monotonicity properties in one-dimensional excited random walks}}, volume = {19}, year = {2013} }

@article{pPPBI, author = {Peterson, Jonathon}, coden = {AMMYAE}, doi = {10.4169/amer.math.monthly.120.06.558}, fjournal = {American Mathematical Monthly}, issn = {0002-9890}, journal = {Amer. Math. Monthly}, mrclass = {05-XX (60-XX)}, mrnumber = {3063121}, number = {6}, pages = {558--562}, title = {{A {P}robabilistic {P}roof of a {B}inomial {I}dentity}}, url = {http://dx.doi.org/10.4169/amer.math.monthly.120.06.558}, volume = {120}, year = {2013}, eprint = {1606.03545} }

@article{psWQLTn, abstract = {We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $\kappa>0$ that determines the fluctuations of the process. When $0<\kappa<2$, the averaged distributions of the hitting times of the random walk converge to a $\kappa$-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times -- viewed as a random probability measure -- converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.}, author = {Peterson, Jonathon and Samorodnitsky, Gennady}, doi = {10.1214/11-AIHP474}, eprint = {1011.6366}, journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.}, number = {3}, pages = {722--752}, title = {{Weak quenched limiting distributions for transient one-dimensional random walk in a random environment}}, url = {http://projecteuclid.org/euclid.aihp/1372772642}, volume = {49}, year = {2013} }

@article{pLDPStrip, author = {Peterson, Jonathon}, eprint = {1302.0888}, fjournal = {ALEA. Latin American Journal of Probability and Mathematical Statistics}, journal = {ALEA Lat. Am. J. Probab. Math. Stat.}, number = {1}, pages = {1--41}, title = {{Large deviations for random walks in a random environment on a strip}}, url = {http://alea.impa.br/articles/v11/11-01.pdf}, volume = {11}, year = {2014} }

@article{pSTRWRE, abstract = {A transient stochastic process is considered strongly transient if conditioned on returning to the starting location, the expected time it takes to return the the starting location is finite. We characterize strong transience for a one-dimensional random walk in a random environment. We show that under the quenched measure transience is equivalent to strong transience, while under the averaged measure strong transience is equivalent to ballisticity (transience with non zero limiting speed).}, author = {Peterson, Jonathon}, doi = {10.1214/ECP.v20-4352}, eprint = {1506.03048}, fjournal = {Electronic Communications in Probability}, issn = {1083-589X}, journal = {Electron. Commun. Probab.}, keywords = {random walk in random environment; strong transience}, pages = {no. 67, 1--10}, title = {{Strong transience of one-dimensional random walk in a random environment}}, url = {http://ecp.ejpecp.org/article/view/4352}, volume = {20}, year = {2015} }

@article{pESDERW, author = {Peterson, Jonathon}, doi = {10.1016/j.spa.2014.09.017}, eprint = {1312.4983}, fjournal = {Stochastic Processes and their Applications}, issn = {0304-4149}, journal = {Stochastic Process. Appl.}, mrclass = {60K35 (60F10 60K37)}, mrnumber = {3293290}, number = {2}, pages = {458--481}, title = {{Extreme slowdowns for one-dimensional excited random walks}}, url = {http://dx.doi.org/10.1016/j.spa.2014.09.017}, volume = {125}, year = {2015} }

@article{apQSA, author = {Ahn, Sung Won and Peterson, Jonathon}, doi = {10.1214/16-EJP4529}, eprint = {1509.00445}, fjournal = {Electronic Journal of Probability}, journal = {Electron. J. Probab.}, pages = {1--27}, pno = {16}, publisher = {The Institute of Mathematical Statistics and the Bernoulli Society}, title = {{Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment}}, url = {http://dx.doi.org/10.1214/16-EJP4529}, volume = {21}, year = {2016} }

@article{kpFLLRERW, abstract = {We consider one-dimensional excited random walks (ERWs) with periodic cookie stacks in the recurrent regime. We prove functional limit theorems for these walks which extend the previous results of D. Dolgopyat and E. Kosygina for excited random walks with "boundedly many cookies per site." In particular, in the non-boundary recurrent case the rescaled excited random walk converges in the standard Skorokhod topology to a Brownian motion perturbed at its extrema (BMPE). While BMPE is a natural limiting object for excited random walks with boundedly many cookies per site, it is far from obvious why the same should be true for our model which allows for infinitely many "cookies" at each site. Moreover, a BMPE has two parameters $\alpha,\beta<1$ and the scaling limits in this paper cover a larger variety of choices for $\alpha$ and $\beta$ than can be obtained for ERWs with boundedly many cookies per site.}, author = {Kosygina, Elena and Peterson, Jonathon}, doi = {10.1214/16-EJP14}, eprint = {1604.03153}, fjournal = {Electronic Journal of Probability}, issn = {1083-6489}, journal = {Electron. J. Probab.}, pages = {1--24}, pno = {70}, sici = {1083-6489(2016)21:70<1:FLLFRE>2.0.CO;2-O}, title = {{Functional limit laws for recurrent excited random walks with periodic cookie stacks}}, url = {http://dx.doi.org/10.1214/16-EJP14}, volume = {21}, year = {2016} }

@article{kpERWMCS, abstract = {We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $\omega_x(j)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $j$ to $x$. The collection $(\omega_x(j))_{x\in\mathbb{Z},n\ge 0}$ is sometimes called the "cookie environment" due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probabilities according to the "strength" of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie "strengths" within the stack $(\omega_x(j))_{j\ge 0}$ at site $x$ follow a finite state Markov chain. Thus, the environment at each site is dynamic, but it evolves according to the local time of the walk at each site rather than the original random walk time. The model admits two different regimes, critical or non-critical, depending on whether the expected probability to jump to the right (or left) under the invariant measure for the Markov chain is equal to $1/2$ or not. We show that in the non-critical regime the walk is always transient, has non-zero linear speed, and satisfies the classical central limit theorem. The critical regime allows for a much more diverse behavior. We give necessary and sufficient conditions for recurrence/transience and ballisticity of the walk in the critical regime as well as a complete characterization of limit laws under the averaged measure in the transient case. The setting considered in this paper generalizes the previously studied model with periodic cookie stacks. Our results on ballisticity and limit theorems are new even for the periodic model.}, author = {Kosygina, Elena and Peterson, Jonathon}, doi = {10.1214/16-AIHP761}, fjournal = {Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques}, journal = {Ann. Inst. H. Poincar\'e Probab. Statist.}, number = {3}, pages = {1458--1497}, publisher = {Institut Henri PoincarĂ©}, title = {Excited random walks with {M}arkovian cookie stacks}, url = {http://dx.doi.org/10.1214/16-AIHP761}, volume = {53}, year = {2017}, eprint = {1504.06280} }

@misc{jpHDLRWRE, abstract = {We consider a system of independent random walks in a common random environment. Previously, a hydrodynamic limit for the system of RWRE was proved under the assumption that the random walks were transient with positive speed. In this paper we instead consider the case where the random walks are transient but with a sublinear speed of the order $n^\kappa$ for some $\kappa \in (0,1)$ and prove a quenched hydrodynamic limit for the system of random walks with time scaled by $n^{1/\kappa}$ and space scaled by $n$. The most interesting feature of the hydrodynamic limit is that the influence of the environment does not average out under the hydrodynamic scaling; that is, the asymptotic particle density depends on the specific environment chosen. The hydrodynamic limit for the system of RWRE is obtained by first proving a hydrodynamic limit for a system of independent particles in a directed trap environment.}, archiveprefix = {arXiv}, author = {Jara, Milton and Peterson, Jonathon}, comment = {published = 2014-10-17T19:43:20Z, updated = 2014-10-17T19:43:20Z, 48 pages, 1 figure}, eprint = {1410.4832}, month = oct, note = {To appear in \emph{Ann. Inst. Henri Poincare{\'e} Probab. Stat.}}, title = {{Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment}}, x-fetchedfrom = {arXiv.org}, year = {2014} }

@misc{dpERWNNNJ, abstract = {Let $W$ be an integer valued random variable satisfying $E[W] =: \delta \geq 0$ and $P(W<0)>0$, and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer $x\in \mathbb{Z}$ the size of the next step is an independent random variable with the same distribution as $W$. We show that this self-interacting random walk is recurrent if $\delta\leq 1$ and transient if $\delta>1$. This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-nearest neighbor jumps.}, archiveprefix = {arXiv}, author = {Davis, Burgess and Peterson, Jonathon}, comment = {published = 2015-04-20T17:01:24Z, updated = 2015-04-20T17:01:24Z}, eprint = {1504.05124}, month = apr, note = {To appear in \emph{J. Theoret. Probab.}}, primaryclass = {math.PR}, title = {{Excited random walks with non-nearest neighbor steps}}, x-fetchedfrom = {arXiv.org}, year = {2015} }

@misc{apQCLTrates, abstract = {Unlike classical simple random walks, one-dimensional random walks in random environments (RWRE) are known to have a wide array of potential limiting distributions. Under certain assumptions, however, it is known that CLT-like limiting distributions hold for the walk under both the quenched and averaged measures. We give upper bounds on the rates of convergence for the quenched central limit theorems for both the hitting time and position of the RWRE with polynomial rates of convergence that depend on the distribution on environments.}, archiveprefix = {arXiv}, author = {Ahn, Sung Won and Peterson, Jonathon}, comment = {published = 2017-04-10T19:12:39Z, updated = 2017-04-10T19:12:39Z, 22 pages, 1 figure}, eprint = {1704.03020v1}, month = apr, primaryclass = {math.PR}, title = {{Quenched central limit theorem rates of convergence for one-dimensional random walks in random environments}}, url = {http://arxiv.org/abs/1704.03020v1; http://arxiv.org/pdf/1704.03020v1}, x-fetchedfrom = {arXiv.org}, year = {2017} }

@misc{prime2016ERW, abstract = {Excited random walks (ERWs) are a self-interacting non-Markovian random walk in which the future behavior of the walk is influenced by the number of times the walk has previously visited its current site. We study the speed of the walk, defined as $V = \lim_{n \rightarrow \infty} \frac{X_n}{n}$ where $X_n$ is the state of the walk at time $n$. While results exist that indicate when the speed is non-zero, there exists no explicit formula for the speed. It is difficult to solve for the speed directly due to complex dependencies in the walk since the next step of the walker depends on how many times the walker has reached the current site. We derive the first non-trivial upper and lower bounds for the speed of the walk. In certain cases these upper and lower bounds are remarkably close together.}, archiveprefix = {arXiv}, author = {Bossen, Erin and Kidd, Brian and Levin, Owen and Peterson, Jonathon and Smith, Jacob and Stangl, Kevin}, comment = {published = 2017-07-10T17:55:49Z, updated = 2017-07-10T17:55:49Z, 11 pages, 3 figures}, eprint = {1707.02969v1}, month = jul, primaryclass = {math.PR}, title = {Upper and Lower Bounds on the Speed of a One Dimensional Excited Random Walk}, url = {http://arxiv.org/abs/1707.02969v1; http://arxiv.org/pdf/1707.02969v1}, x-fetchedfrom = {arXiv.org}, year = {2017} }

@misc{gpBERP, abstract = {We prove Berry-Esseen type rates of convergence for central limit theorems of regenerative processes which generalize previous results of Bolthausen under weaker moment assumptions. As an application we give rates of convergence of the central limit theorem under the annealed measure for certain ballistic random walks in random environments.}, archiveprefix = {arXiv}, author = {Xiaoqin Guo and Jonathon Peterson}, comment = {published = 2017-08-23T19:39:02Z, updated = 2017-08-23T19:39:02Z}, eprint = {1708.07162v1}, month = aug, primaryclass = {math.PR}, title = {{Berry-Esseen estimates for regenerative processes}}, url = {http://arxiv.org/abs/1708.07162v1; http://arxiv.org/pdf/1708.07162v1}, x-fetchedfrom = {arXiv.org}, year = {2017} }

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