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http://hdl.handle.net/1813/525
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http://hdl.handle.net/1813/525
Tauberian Theory for Multivariate Regularly Varying Distributions with Application to Preferential Attachment Networks
http://hdl.handle.net/1813/36712
Title: Tauberian Theory for Multivariate Regularly Varying Distributions with Application to Preferential Attachment Networks
Authors: Resnick, Sidney; Samorodnitsky, Gennady
Abstract: Abel-Tauberian theorems relate power
law behavior of distributions and their transforms. We formulate and
prove a multivariate version for non-standard regularly varying
measures on R_+^p and then apply it to
prove that the joint distribution of in- and out-degree in a directed edge
preferential attachement model has jointly regularly varying
tails.Wed, 25 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1813/367122014-06-25T00:00:00ZNonstandard regular variation of the in-degree and the out-degree in the preferential attachement model
http://hdl.handle.net/1813/36711
Title: Nonstandard regular variation of the in-degree and the out-degree in the preferential attachement model
Authors: Samorodnitsky, Gennady; Resnick, Sidney; Towsley, Don; Davis, Richard; Willis, Amy; Wan, Phyllis
Abstract: For the directed edge preferential attachment network growth model
studied by Bollobas et al. (2003) and
Krapivsky and Redner (2001), we prove that the joint distribution of
in-degree and
out-degree
has jointly regularly varying
tails.
Typically the marginal tails of the in-degree distribution and the out-degree
distribution have different regular variation indices and so the joint
regular variation is non-standard.
Only marginal regular variation has been
previously established for this distribution in the cases where the
marginal tail indices are different.Wed, 25 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1813/367112014-06-25T00:00:00ZGeneral inverse problems for regular variation
http://hdl.handle.net/1813/34411
Title: General inverse problems for regular variation
Authors: Damek, Ewa; Mikosch, Thomas; Rosinski, Jan; Samorodnitsky, Gennady
Abstract: Regular variation of distributional tails is known to be preserved by
various linear transformations of some random structures.
An inverse problem for regular
variation aims at understanding whether the regular variation of a
transformed random object is caused by regular variation of components
of the original random structure. In this paper we build up on previous
work and derive results in the multivariate case and in
situations where regular variation is
not restricted to one particular direction or quadrant.Wed, 02 Oct 2013 00:00:00 GMThttp://hdl.handle.net/1813/344112013-10-02T00:00:00ZStock Optimization in Emergency Resupply Networks under Stuttering Poisson Demand
http://hdl.handle.net/1813/33186
Title: Stock Optimization in Emergency Resupply Networks under Stuttering Poisson Demand
Authors: Chen, J; Jackson, P.L.; Muckstadt, J
Abstract: We consider a network in which field stocking locations (FSLs) manage multiple parts according
to an (S-1,S) policy. Demand processes for the parts are assumed to be independent
stuttering Poisson processes. Regular replenishments to an FSL occur from a regional stocking
location (RSL) that has an unlimited supply of each part type. Demand in excess
of supply at an FSL is routed to an emergency stocking location (ESL), which also employs
an (S-1,S) policy to manage its inventory. Demand in excess of supply at the ESL is backordered.
Lead time from the ESL to each FSL is assumed to be negligible compared to the
RSL-ESL resupply time. In companion papers we have shown how to approximate the joint
probability distributions of units on hand, units in regular resupply, and units in emergency
resupply. In this paper, we focus on the problem of determining the stock levels at the FSLs
and ESL across all part numbers that minimize backorder, and emergency resupply costs
subject to an inventory investment budget constraint. The problem is shown to be a nonconvex integer programming problem, and we explore a collection of heuristics for solving
the optimization problem.Mon, 01 Apr 2013 00:00:00 GMThttp://hdl.handle.net/1813/331862013-04-01T00:00:00ZCalculation of ruin probabilities for a dense class of heavy tailed distributions
http://hdl.handle.net/1813/31540
Title: Calculation of ruin probabilities for a dense class of heavy tailed distributions
Authors: Bladt, Mogens; Nielsen, Bo Friis; Samorodnitsky, Gennady
Abstract: In this paper we propose a
class of infinite--dimensional phase--type distributions with
finitely many parameters as models for heavy tailed distributions.
The class of finite--dimensional distributions is dense in the class
of distributions on the positive reals and may hence approximate any
such distribution.
We
prove that formulas from renewal theory, and with a particular
attention to ruin probabilities, which are true for common
phase--type distributions also hold true for the
infinite--dimensional case. We provide algorithms for
calculating functionals of interest such as the renewal density and
the ruin probability. It might be of interest to approximate a given
heavy--tailed distribution of some other type by a distribution from
the class of infinite--dimensional phase--type distributions and to
this end we provide a calibration procedure which works for the
approximation of distributions with a slowly varying tail. An
example from risk theory, comparing ruin probabilities for a
classical risk process with Pareto distributed claim sizes, is
presented and exact known ruin probabilities for the Pareto case are
compared to the ones obtained by approximating by an
infinite--dimensional hyper--exponential distribution.Tue, 05 Mar 2013 00:00:00 GMThttp://hdl.handle.net/1813/315402013-03-05T00:00:00ZMultivariate tail measure and the estimation of CoVar
http://hdl.handle.net/1813/30442
Title: Multivariate tail measure and the estimation of CoVar
Authors: Nguyen, Tilo; Samorodnitsky, Gennady
Abstract: The quality of estimation of multivariate tails depends significantly
on the portion of the sample included in the estimation. A simple
approach involving sequential statistical testing is proposed in order
to select which observations should be used for estimation of the tail
and spectral measures. We prove that the estimator is consistent. We
test the proposed method on simulated data, and subsequently apply it
to analyze CoVar for stock and index returns.Tue, 09 Oct 2012 00:00:00 GMThttp://hdl.handle.net/1813/304422012-10-09T00:00:00ZFunctional Central Limit Theorem for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows
http://hdl.handle.net/1813/29996
Title: Functional Central Limit Theorem for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows
Authors: Owada, Takashi; Samorodnitsky, Gennady
Abstract: We establish a new class of functional central limit theorems for
partial sum of certain symmetric stationary infinitely divisible processes with
regularly varying Levy measures. The limit process is a new class of
symmetric stable self-similar processes with stationary increments,
that coincides on a part of its parameter space with a previously
described process. The normalizing sequence and the limiting process
are determined by the ergodic theoretical properties of the flow
underlying the integral representation of the process. These
properties can be interpreted as determining how long is the memory of
the stationary infinitely divisible process. We also
establish functional convergence, in a strong distributional sense,
for conservative pointwise dual ergodic maps preserving an infinite
measure.Tue, 18 Sep 2012 00:00:00 GMThttp://hdl.handle.net/1813/299962012-09-18T00:00:00ZIntrinsic location functionals of stationary processes
http://hdl.handle.net/1813/29085
Title: Intrinsic location functionals of stationary processes
Authors: Samorodnitsky, Gennady; Shen, Yi
Abstract: We consider a large family of measurable functionals of the sample
path of a stochastic process over compact intervals (including first
hitting times,
leftmost location of the supremum, etc.) we call intrinsic location
functionals. Despite the large variety of these functionals and their
different nature, we show that for stationary processes
the distribution of any intrinsic location functional over an interval
is absolute continuous in the interior of the interval, and the
density functions always have a version satisfying
the same total variation constraints. Conversely, these total
variation constraints are shown to actually characterize stationarity
of the underlying stochastic process. We also show that
the possible distributions of the intrinsic location functionals over
an interval form a weakly closed convex set and describe its extreme
points, and present applications of this description.Thu, 21 Jun 2012 00:00:00 GMThttp://hdl.handle.net/1813/290852012-06-21T00:00:00ZOn the existence of paths between points in high level excursion sets of Gaussian random fields
http://hdl.handle.net/1813/28637
Title: On the existence of paths between points in high level excursion sets of Gaussian random fields
Authors: Adler, Robert; Moldavskaya, Elina; Samorodnitsky, Gennady
Abstract: The structure of Gaussian random fields over high levels is a well researched and
well understood area, particularly if the field is smooth. However, the question as to whether or not two or more points
which lie in an
excursion set belong to the same connected component has constantly eluded analysis. We study this problem from the point of view of large deviations, finding the asymptotic
probabilities that two such points are connected by a path laying within the excursion set,
and so belong to the same component. In addition, we obtain a characterization and descriptions of the
most likely paths, given that one exists.Tue, 27 Mar 2012 00:00:00 GMThttp://hdl.handle.net/1813/286372012-03-27T00:00:00ZFractional moments of solutions to stochastic recurrence equations
http://hdl.handle.net/1813/28598
Title: Fractional moments of solutions to stochastic recurrence equations
Authors: Mikosch, Thomas; Samorodnitsky, Gennady; Tafakori, Laleh
Abstract: In this paper we study the fractional moments of the stationary
solution to a stochastic recursion.
We derive recursive formulas for the fractional moments of the solution. Special attention is given to the case when
the additive term has an Erlang distribution. We provide various approximations to the
moments and show their performance in a small numerical studyTue, 13 Mar 2012 00:00:00 GMThttp://hdl.handle.net/1813/285982012-03-13T00:00:00Z