10,459 research outputs found
Indicator fractional stable motions
Using the framework of random walks in random scenery, Cohen and
Samorodnitsky (2006) introduced a family of symmetric -stable motions
called local time fractional stable motions. When , these processes
are precisely fractional Brownian motions with . Motivated by random
walks in alternating scenery, we find a "complementary" family of symmetric
-stable motions which we call indicator fractional stable motions.
These processes are complementary to local time fractional stable motions in
that when , one gets fractional Brownian motions with .Comment: 11 pages, final version as accepted in Electronic Communications in
Probabilit
Random walks at random times: Convergence to iterated L\'{e}vy motion, fractional stable motions, and other self-similar processes
For a random walk defined for a doubly infinite sequence of times, we let the
time parameter itself be an integer-valued process, and call the orginal
process a random walk at random time. We find the scaling limit which
generalizes the so-called iterated Brownian motion. Khoshnevisan and Lewis
[Ann. Appl. Probab. 9 (1999) 629-667] suggested "the existence of a form of
measure-theoretic duality" between iterated Brownian motion and a Brownian
motion in random scenery. We show that a random walk at random time can be
considered a random walk in "alternating" scenery, thus hinting at a mechanism
behind this duality. Following Cohen and Samorodnitsky [Ann. Appl. Probab. 16
(2006) 1432-1461], we also consider alternating random reward schema associated
to random walks at random times. Whereas random reward schema scale to local
time fractional stable motions, we show that the alternating random reward
schema scale to indicator fractional stable motions. Finally, we show that one
may recursively "subordinate" random time processes to get new local time and
indicator fractional stable motions and new stable processes in random scenery
or at random times. When , the fractional stable motions given by the
recursion are fractional Brownian motions with dyadic . Also, we see
that "un-subordinating" via a time-change allows one to, in some sense, extract
Brownian motion from fractional Brownian motions with .Comment: Published in at http://dx.doi.org/10.1214/12-AOP770 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Two Phase Transitions for the Contact Process on Small Worlds
In our version of Watts and Strogatz's small world model, space is a
d-dimensional torus in which each individual has in addition exactly one
long-range neighbor chosen at random from the grid. This modification is
natural if one thinks of a town where an individual's interactions at school,
at work, or in social situations introduces long-range connections. However,
this change dramatically alters the behavior of the contact process, producing
two phase transitions. We establish this by relating the small world to an
infinite "big world" graph where the contact process behavior is similar to the
contact process on a tree.Comment: 24 pages, 6 figures. We have rewritten the phase transition in terms
of two parameters and have made improvements to our original result
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