scholarly journals On the asymptotic of the maximal weighted increment of a random walk with regularly varying jumps: the boundary case

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Alfredas Račkauskas ◽  
Charles Suquet
Keyword(s):  
Random Walk ◽  
Boundary Case ◽  
Regularly Varying

scholarly journals Markov Tail Chains

2014 ◽  
Vol 51 (4) ◽  
pp. 1133-1153 ◽  
Author(s):  
A. Janssen ◽  
J. Segers
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Marginal Distribution ◽  
Extreme Event ◽  
Asymptotically Linear ◽  
The Past ◽  
Linear Behavior ◽  
Regularly Varying ◽  
Theory Yield ◽  
Underlying Processes

The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions inRd. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

Phase transition in one-dimensional random walk with partially reflecting boundaries

1985 ◽  
Vol 17 (03) ◽  
pp. 594-606 ◽  
Author(s):  
Ora E. Percus
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Boundary Case ◽  
Reflecting Boundaries ◽  
The Mean ◽  
The One ◽  
The Relationship ◽  
Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p). The probability distribution (Pn (m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of P n(m) result, depending on the relationship between transition probability and the reflection probability.

Phase transition in one-dimensional random walk with partially reflecting boundaries

1985 ◽  
Vol 17 (3) ◽  
pp. 594-606 ◽  
Author(s):  
Ora E. Percus
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Boundary Case ◽  
Reflecting Boundaries ◽  
The Mean ◽  
The One ◽  
The Relationship ◽  
Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

Trinomial random walk with one or two imperfect absorbing barriers

2008 ◽  
Vol 58 (3) ◽  
Author(s):  
M. El-Shehawey
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Transition Probabilities ◽  
Boundary Case ◽  
The Mean ◽  
The One ◽  
Annihilation Probability ◽  
The Relationship ◽  
Two Barriers ◽  
Asymptotic Forms

AbstractTrinomial random walk, with one or two barriers, on the non-negative integers is discussed. At the barriers, the particle is either annihilated or reflects back to the system with respective probabilities 1 − ρ, ρ at the origin and 1 − ω, ω at L, 0 ≤ ρ,ω ≤ 1. Theoretical formulae are given for the probability distribution, its generating function as well as the mean of the time taken before absorption. In the one-boundary case, very qualitatively different asymptotic forms for the result, depending on the relationship between transition probabilities and the annihilation probability, are obtained.

scholarly journals A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk

2015 ◽  
Vol 47 (03) ◽  
pp. 741-760
Author(s):  
Xinxin Chen
Keyword(s):  
Fixed Point ◽  
Random Walk ◽  
Branching Random Walk ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Boundary Case ◽  
Necessary And Sufficient ◽  
Smoothing Transformation

We consider a branching random walk. Biggins and Kyprianou (2004) proved that, in the boundary case, the associated derivative martingale converges almost surely to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In this paper, we give a necessary and sufficient condition for the nontriviality of the limit in this boundary case.

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