The Experts below are selected from a list of 264 Experts worldwide ranked by ideXlab platform
Kôhei Uchiyama - One of the best experts on this subject based on the ideXlab platform.
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Scaling Limits of Random Walk Bridges Conditioned to Avoid a Finite Set
Journal of Theoretical Probability, 2019Co-Authors: Kôhei UchiyamaAbstract:This paper concerns a scaling limit of a One-Dimensional Random Walk $$S^x_n$$ S n x started from x on the integer lattice conditioned to avoid a non-empty finite set A , the Random Walk being assumed to be irreducible and have zero mean. Suppose the variance $$\sigma ^2$$ σ 2 of the increment law is finite. Given positive constants b , c and T , we consider the scaled process $$S^{b_N}_{[tN]}/\sigma \sqrt{N}$$ S [ t N ] b N / σ N , $$0\le t \le T$$ 0 ≤ t ≤ T , started from a point $$b_N \approx b\sqrt{N}$$ b N ≈ b N conditioned to arrive at another point $$\approx -\,c\sqrt{N}$$ ≈ - c N at $$t=T$$ t = T and avoid A in between and discuss the functional limit of it as $$N\rightarrow \infty $$ N → ∞ . We show that it converges in law to a continuous process if $$E[|S_1|^3; S_1
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One dimensional Random Walks killed on a finite set
Stochastic Processes and their Applications, 2017Co-Authors: Kôhei UchiyamaAbstract:We study the transition probability, say pAn(x,y), of a One-Dimensional Random Walk on the integer lattice killed when entering into a non-empty finite set A. The Random Walk is assumed to be irreducible and have zero mean and a finite variance σ2. We show that pAn(x,y) behaves like [gA+(x)gA+(y)+gA−(x)gA−(y)](σ2/2n)pn(y−x) uniformly in the regime characterized by the conditions ∣x∣∨∣y∣=O(n) and ∣x∣∧∣y∣=o(n) generally if xy>0 and under a mild additional assumption about the Walk if xy
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One dimensional Random Walk killed on a finite set
arXiv: Probability, 2016Co-Authors: Kôhei UchiyamaAbstract:We study the transition probability, say $p_A^n(x,y)$, of a One-Dimensional Random Walk on the integer lattice killed when entering into a non-empty finite set $A$. The Random Walk is assumed to be irreducible and have zero mean and a finite variance $\sigma^2$. We derive the asymptotic form of $p_A^n(x, y)$ for large $n$ valid uniformly in the regime characterized by the conditions $|x|\vee |y| =O(\sqrt n)$ and $|x|\wedge |y|= o(\sqrt n)$, in which $p^A_t({\bf x},{\bf y})$ behaves for large $n$ like $[g_A^{+}(x)\hat g_{A}^{\,+}(y) + g_A^-(x)\hat g_{A}^{\,-}(y)] (\sigma^{2}/2n) p^n(y-x)$. Here $p^n(y-x)$ is the transition kernel of the Random Walk (without killing); $g^\pm_A$ are the Green functions for the "exterior" of $A$ with "pole at $\pm \infty$" normalized so that $g^\pm_A(x) \sim 2|x|/\sigma^2$ as $x \to \pm\infty$; and $\hat g_A^{\, \pm}$ are the corresponding Green functions for the time-reversed Walk.
Julien Brémont - One of the best experts on this subject based on the ideXlab platform.
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Random Walk IN QUASI-PERIODIC Random ENVIRONMENT
Stochastics and Dynamics, 2009Co-Authors: Julien BrémontAbstract:We consider a One-Dimensional Random Walk with finite range in a Random medium described by an ergodic translation on a torus. For regular data and under a Diophantine condition on the translation, we prove a central limit theorem with deterministic centering.
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Random Walks in Random medium on ℤ and Lyapunov spectrum
Annales de l'institut Henri Poincare (B) Probability and Statistics, 2004Co-Authors: Julien BrémontAbstract:We consider a One-Dimensional Random Walk with bounded steps in a stationary and ergodic Random medium. We show that the algebraic structure of the Random Walk is given by geometrical invariants related to the description of a space of harmonic functions. We then prove a recurrence criterion similar to Key's Theorem [E.S. Key, Ann. Probab. 12 (2) (1984) 529] in terms of the sign of an intermediate Lyapunov exponent of a Random matrix. We show that this exponent is simple and we relate it to the dominant exponents of two non-negative matrices associated to the Random Walks of left and right records. We also give an algorithm to compute that exponent. In a last part, we deduce from [J. Brémont, Ann. Probab. 30 (3) (2002) 1266] that the Law of Large Numbers is always valid. © 2004 Elsevier SAS. All rights reserved.
Jonathon Peterson - One of the best experts on this subject based on the ideXlab platform.
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Quenched limits for transient, zero speed One-Dimensional Random Walk in Random environment.
Annals of Probability, 2020Co-Authors: Jonathon Peterson, Ofer ZeitouniAbstract:We consider a nearest-neighbor, one dimensional Random Walk {X n } n≥0 in a Random i.i.d. environment, in the regime where the Walk is transient but with zero speed, so that X n is of order n S for some s 0 and → 0 for x ≤ 0 (a spread out regime).
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Strong transience of One-Dimensional Random Walk in a Random environment
Electronic Communications in Probability, 2015Co-Authors: Jonathon PetersonAbstract: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).
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Oscillations of quenched slowdown asymptotics for ballistic One-Dimensional Random Walk in a Random environment
arXiv: Probability, 2015Co-Authors: Jonathon PetersonAbstract:We consider a one dimensional Random Walk in a Random environment (RWRE) with a positive speed $\lim_{n\to\infty}\frac{X_n}{n}=v_\alpha>0$. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_\omega(X_n 1$. More precisely, they showed that $n^{-\gamma} \log P_\omega( X_n 1-1/s$ or $\gamma < 1-1/s$. In this paper, we improve on this by showing that $n^{-1+1/s} \log P_\omega( X_n < x n)$ oscillates between $0$ and $-\infty$, almost surely. This had previously been shown by Gantert only in a very special case of Random environments.
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Weak quenched limiting distributions for transient One-Dimensional Random Walk in a Random environment
Annales De L Institut Henri Poincare-probabilites Et Statistiques, 2013Co-Authors: Jonathon Peterson, Gennady SamorodnitskyAbstract:J. Peterson was partially supported by National Science Foundation grant DMS-0802942. G. Samorodnitsky was partially supported by ARO grant W911NF-10-1-0289 and NSF grant DMS-1005903 at Cornell University
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Weak quenched limiting distributions for transient One-Dimensional Random Walk in a Random environment
arXiv: Probability, 2010Co-Authors: Jonathon Peterson, Gennady SamorodnitskyAbstract: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
Guo'ai Xu - One of the best experts on this subject based on the ideXlab platform.
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CCIS - Construction of distributed LDoS attack based on One-Dimensional Random Walk algorithm
2012 IEEE 2nd International Conference on Cloud Computing and Intelligence Systems, 2012Co-Authors: Miao Zhang, Guo'ai XuAbstract:This paper designs a distributed low-rate DoS (LDoS) attack using Random Walk algorithm; Random Walk algorithm can be used in generating power-law distribution which complies with normal network behavior feature. In this attack pattern, the behavior of each distributed attack flows are normal and nonperiodic, while the superposed effect has the same damage degree as traditional square wave LDoS attacks. As the distributed attack flow satisfies the network traffic behavior feature, it is more difficult to detect and trace attack sources. The study on the feature of this attack provides reference for distributed low-rate DoS attack countermeasures.
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Construction of distributed LDoS attack based on One-Dimensional Random Walk algorithm
2012 IEEE 2nd International Conference on Cloud Computing and Intelligence Systems, 2012Co-Authors: Miao Zhang, Guo'ai XuAbstract:This paper designs a distributed low-rate DoS (LDoS) attack using Random Walk algorithm; Random Walk algorithm can be used in generating power-law distribution which complies with normal network behavior feature. In this attack pattern, the behavior of each distributed attack flows are normal and nonperiodic, while the superposed effect has the same damage degree as traditional square wave LDoS attacks. As the distributed attack flow satisfies the network traffic behavior feature, it is more difficult to detect and trace attack sources. The study on the feature of this attack provides reference for distributed low-rate DoS attack countermeasures.
Bruno Schapira - One of the best experts on this subject based on the ideXlab platform.
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The quenched limiting distributions of a One-Dimensional Random Walk in Random scenery
arXiv: Probability, 2013Co-Authors: Nadine Guillotin-plantard, Yueyun Hu, Bruno SchapiraAbstract:For a One-Dimensional Random Walk in Random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of the iterated logarithm. As a consequence, conditioned on the Random scenery, the One-Dimensional RWRS does not converge in law, in contrast with the multi-dimensional case.
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The quenched limiting distributions of a One-Dimensional Random Walk in Random scenery
Electronic Communications in Probability, 2013Co-Authors: Nadine Guillotin-plantard, Yueyun Hu, Bruno SchapiraAbstract:For a One-Dimensional Random Walk in Random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of iterated logarithm. As a consequence, conditioned on the Random scenery, the one dimensional RWRS does not converge in law, in contrast with the multi-dimensional case.
