55 research outputs found
The almost-sure asymptotic behavior of the solution to the stochastic heat equation with L\'evy noise
We examine the almost-sure asymptotics of the solution to the stochastic heat
equation driven by a L\'evy space-time white noise. When a spatial point is
fixed and time tends to infinity, we show that the solution develops unusually
high peaks over short time intervals, even in the case of additive noise, which
leads to a breakdown of an intuitively expected strong law of large numbers.
More precisely, if we normalize the solution by an increasing nonnegative
function, we either obtain convergence to , or the limit superior and/or
inferior will be infinite. A detailed analysis of the jumps further reveals
that the strong law of large numbers can be recovered on discrete sequences of
time points increasing to infinity. This leads to a necessary and sufficient
condition that depends on the L\'evy measure of the noise and the growth and
concentration properties of the sequence at the same time. Finally, we show
that our results generalize to the stochastic heat equation with a
multiplicative nonlinearity that is bounded away from zero and infinity.Comment: Forthcoming in The Annals of Probabilit
A note on a maximal Bernstein inequality
We show somewhat unexpectedly that whenever a general Bernstein-type maximal
inequality holds for partial sums of a sequence of random variables, a maximal
form of the inequality is also valid.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ304 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The limit distribution of ratios of jumps and sums of jumps of subordinators
Let be a driftless subordinator, and let denote its jump sequence on interval . Put for the -trimmed subordinator.
In this note we characterize under what conditions the limiting distribution of
the ratios and
exist, as or .Comment: 14 page
Couplings and Strong Approximations to Time Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions
We define a time dependent empirical process based on i.i.d.~fractional
Brownian motions and establish Gaussian couplings and strong approximations to
it by Gaussian processes. They lead to functional laws of the iterated
logarithm for this process.Comment: To appear in the Journal of Theoretical Probability. 37 pages.
Corrected version. The results on quantile processes are taken out and it
will appear elsewher
Asymptotic behavior of the generalized St. Petersburg sum conditioned on its maximum
In this paper, we revisit the classical results on the generalized St.
Petersburg sums. We determine the limit distribution of the St. Petersburg sum
conditioning on its maximum, and we analyze how the limit depends on the value
of the maximum. As an application, we obtain an infinite sum representation of
the distribution function of the possible semistable limits. In the
representation, each term corresponds to a given maximum, in particular this
result explains that the semistable behavior is caused by the typical values of
the maximum.Comment: Published at http://dx.doi.org/10.3150/14-BEJ685 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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