281 research outputs found
Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case
As well known, for a supercritical Galton-Watson process whose
offspring distribution has mean , the ratio has a.s. limit,
say . We study tail behaviour of the distributions of and in the
case where has heavy-tailed distribution, that is, \E e^{\lambda
Z_1}=\infty for every . We show how different types of
distributions of lead to different asymptotic behaviour of the tail of
and . We describe the most likely way how large values of the process
occur
Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift
We consider a Markov chain on with asymptotically zero drift and finite
second moments of jumps which is positive recurrent. A power-like asymptotic
behaviour of the invariant tail distribution is proven; such a heavy-tailed
invariant measure happens even if the jumps of the chain are bounded. Our
analysis is based on test functions technique and on construction of a harmonic
function.Comment: 27 page
At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift
In Chapter 2 we introduce a classification of Markov chains with
asymptotically zero drift, which relies on relations between first and second
moments of jumps. We construct an abstract Lyapunov functions which looks
similar to functions which characterise the behaviour of diffusions with
similar drift and diffusion coefficient.
Chapter 3 is devoted to the limiting behaviour of transient chains. Here we
prove converges to and normal distribution which generalises papers by
Lamperti, Kersting and Klebaner. We also determine the asymptotic behaviour of
the cumulative renewal function.
In Chapter 4 we introduce a general strategy of change of measure for Markov
chains with asymptotically zero drift. This is the most important ingredient in
our approach to recurrent chains.
Chapter 5 is devoted to the study of the limiting behaviour of recurrent
chains with the drift proportional to . We derive asymptotics for a
stationary measure and determine the tail behaviour of recurrence times. All
these asymptotics are of power type.
In Chapter 6 we show that if the drift is of order then moments
of all orders are important for the behaviour of stationary
distributions and pre-limiting tails. Here we obtain Weibull-like asymptotics.
In Chapter 7 we apply our results to different processes, e.g. critical and
near-critical branching processes, risk processes with reserve-dependent
premium rate, random walks conditioned to stay positive and reflected random
walks.
In Chapter 8 we consider asymptotically homogeneous in space Markov chains
for which we derive exponential tail asymptotics
On lower limits and equivalences for distribution tails of randomly stopped sums
For a distribution of a random sum
of i.i.d. random variables with a common
distribution on the half-line , we study the limits of the
ratios of tails as (here,
is a counting random variable which does not depend on ). We
also consider applications of the results obtained to random walks, compound
Poisson distributions, infinitely divisible laws, and subcritical branching
processes.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ111 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Visualizing the strain evolution during the indentation of colloidal glasses
We use an analogue of nanoindentation on a colloidal glass to elucidate the
incipient plastic deformation of glasses. By tracking the motion of the
individual particles in three dimensions, we visualize the strain field and
glass structure during the emerging deformation. At the onset of flow, we
observe a power-law distribution of strain indicating strongly correlated
deformation, and reflecting a critical state of the glass. At later stages, the
strain acquires a Gaussian distribution, indicating that plastic events become
uncorrelated. Investigation of the glass structure using both static and
dynamic measures shows a weak correlation between the structure and the
emerging strain distribution. These results indicate that the onset of
plasticity is governed by strong power-law correlations of strain, weakly
biased by the heterogeneous glass structure.Comment: 13 pages, 8 figure
PROBABILITY METHODS FOR ASSESSING FINANCIAL RISKS FOR ENTERPRISES
The paper reviews methods for risk assessment which could be employed in the financial management of enterprises. The set of methods proposed renders it possible to assess the impact which different risk factors upon the efficiency of implemented projects and the effect of antirisk measures on the financial performance of companies and thus identify the most efficient measures according to the criterion selected for project evaluation
Asymptotics of randomly stopped sums in the presence of heavy tails
We study conditions under which as , where is a sum of
random size and is a maximum of partial sums
. Here , , 2, ..., are independent
identically distributed random variables whose common distribution is assumed
to be subexponential. We consider mostly the case where is independent
of the summands; also, in a particular situation, we deal with a stopping time.
Also we consider the case where and where the tail of is
comparable with or heavier than that of , and obtain the asymptotics
as . This case
is of a primary interest in the branching processes.
In addition, we obtain new uniform (in all and ) upper bounds for the
ratio which substantially improve Kesten's bound in the
subclass of subexponential distributions.Comment: 22 page
Renewal Theory for Transient Markov Chains with Asymptotically Zero Drift
We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain in , that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by in the interval is roughly speaking the reciprocal of the drift and tends to infinity as grows. For the first time we present a general approach relying on a diffusion approximation to prove renewal theorems for Markov chains. We apply a martingale type technique and show that the asymptotic behaviour of the renewal measure heavily depends on the rate at which the drift vanishes. The two main cases are distinguished, either the drift of the chain decreases as or much slower than that, say as for some . The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case converges weakly to a -distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for and further normal approximation is available
- …