4,856 research outputs found
Markov chains conditioned never to wait too long at the origin
Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by Ο the first time that the chain, X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (ΟβΊT). We show that there is a weak limit as Tββ in the cases where either the state space is finite or X is transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than Ο and Ο is subexponential
Right inverses of L\'{e}vy processes
We call a right-continuous increasing process a partial right inverse
(PRI) of a given L\'{e}vy process if for at least all in
some random interval of positive length. In this paper, we give a
necessary and sufficient condition for the existence of a PRI in terms of the
L\'{e}vy triplet.Comment: Published in at http://dx.doi.org/10.1214/09-AOP515 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Stochastic bounds for Levy processes
Using the Wiener-Hopf factorization, it is shown that it is possible to bound
the path of an arbitrary Levy process above and below by the paths of two
random walks. These walks have the same step distribution, but different random
starting points. In principle, this allows one to deduce Levy process versions
of many known results about the large-time behavior of random walks. This is
illustrated by establishing a comprehensive theorem about Levy processes which
converge to \infty in probability.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000031
Curve crossing for random walks reflected at their maximum
Let be a random walk reflected in its
maximum. Except in the trivial case when , will pass over a
horizontal boundary of any height in a finite time, with probability 1. We
extend this by giving necessary and sufficient conditions for finiteness of
passage times of above certain curved (power law) boundaries, as well.
The intuition that a degree of heaviness of the negative tail of the
distribution of the increments of is necessary for passage of above
a high level is correct in most, but not all, cases, as we show. Conditions are
also given for the finiteness of the expected passage time of above
linear and square root boundaries.Comment: Published at http://dx.doi.org/10.1214/009117906000000953 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Local asymptotics for the time of first return to the origin of transient random walk
We consider a transient random walk on which is asymptotically stable,
without centering, in a sense which allows different norming for each
component. The paper is devoted to the asymptotics of the probability of the
first return to the origin of such a random walk at time
- β¦