233 research outputs found
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
Small and Large Time Stability of the Time taken for a L\'evy Process to Cross Curved Boundaries
This paper is concerned with the small time behaviour of a L\'{e}vy process
. In particular, we investigate the {\it stabilities} of the times,
\Tstarb(r) and \Tbarb(r), at which , started with , first leaves
the space-time regions (one-sided exit),
or (two-sided exit), , as
r\dto 0. Thus essentially we determine whether or not these passage times
behave like deterministic functions in the sense of different modes of
convergence; specifically convergence in probability, almost surely and in
. In many instances these are seen to be equivalent to relative stability
of the process itself. The analogous large time problem is also discussed
Passage time and fluctuation calculations for subexponential L\'evy processes
We consider the passage time problem for L\'evy processes, emphasising heavy
tailed cases. Results are obtained under quite mild assumptions, namely, drift
to a.s. of the process, possibly at a linear rate (the finite mean
case), but possibly much faster (the infinite mean case), together with
subexponential growth on the positive side. Local and functional versions of
limit distributions are derived for the passage time itself, as well as for the
position of the process just prior to passage, and the overshoot of a high
level. A significant connection is made with extreme value theory via regular
variation or maximum domain of attraction conditions imposed on the positive
tail of the canonical measure, which are shown to be necessary for the kind of
convergence behaviour we are interested in.Comment: Published at http://dx.doi.org/10.3150/15-BEJ700 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Stability of the Exit Time for L\'evy Processes
This paper is concerned with the behaviour of a L\'{e}vy process when it
crosses over a positive level, , starting from 0, both as becomes large
and as becomes small. Our main focus is on the time, , it takes the
process to transit above the level, and in particular, on the {\it stability}
of this passage time; thus, essentially, whether or not behaves
linearly as u\dto 0 or . We also consider conditional stability
of when the process drifts to , a.s. This provides
information relevant to quantities associated with the ruin of an insurance
risk process, which we analyse under a Cram\'er condition
Distributional representations and dominance of a L\'{e}vy process over its maximal jump processes
Distributional identities for a L\'evy process , its quadratic variation
process and its maximal jump processes, are derived, and used to make
"small time" (as ) asymptotic comparisons between them. The
representations are constructed using properties of the underlying Poisson
point process of the jumps of . Apart from providing insight into the
connections between , , and their maximal jump processes, they enable
investigation of a great variety of limiting behaviours. As an application, we
study "self-normalised" versions of , that is, after division by
, or by . Thus, we
obtain necessary and sufficient conditions for
and to converge in probability to 1, or to
, as , so that is either comparable to, or dominates,
its largest jump. The former situation tends to occur when the singularity at 0
of the L\'evy measure of is fairly mild (its tail is slowly varying at 0),
while the latter situation is related to the relative stability or attraction
to normality of at 0 (a steeper singularity at 0). An important component
in the analyses is the way the largest positive and negative jumps interact
with each other. Analogous "large time" (as ) versions of the
results can also be obtained.Comment: Published at http://dx.doi.org/10.3150/15-BEJ731 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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