eCommons Collection:http://hdl.handle.net/1813/39422015-02-25T23:35:16Z2015-02-25T23:35:16ZClimbing down Gaussian peaksAdler, RobertSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/390772015-01-29T06:02:16Z2015-01-28T00:00:00ZTitle: Climbing down Gaussian peaks
Authors: Adler, Robert; Samorodnitsky, Gennady
Abstract: How likely is the high level of a continuous Gaussian random field on
an Euclidean space to
have a ``hole'' of a certain dimension and depth? Questions of this
type are difficult, but in this paper we make progress on questions
shedding new light in existence of holes. How likely is the field to
be above a high level on one compact set (e.g. a sphere) and to be
below a fraction of that level on some other compact set, e.g. at the
center of the corresponding ball? How likely is the field to be below that
fraction of the level anywhere nside the ball? We work on the
level of large deviations.2015-01-28T00:00:00ZNumerical Validation of Fill Rate Estimation Methods for Two- and Three-Demand Class Rationing Policies with One-for-One Replenishment and General Lead Time DistributionsVicil, OguzhanJackson, Peterhttp://hdl.handle.net/1813/390192015-01-09T06:03:23Z2015-01-08T00:00:00ZTitle: Numerical Validation of Fill Rate Estimation Methods for Two- and Three-Demand Class Rationing Policies with One-for-One Replenishment and General Lead Time Distributions
Authors: Vicil, Oguzhan; Jackson, Peter
Abstract: In this report, we conduct numerical simulations of two- and three-demand class inventory threshold rationing systems under one-for-one replenishment policies. The performance metrics of interest are the fill rates of the high priority demand classes (the gold fill rate in the two demand
class system and the platinum and gold fill rates in the three-demand class system).
Our main interest is in the sensitivity of these fill rates to the form of the replenishment lead time probability distribution and the resulting quality of approximation methods used to estimate
these fill rates. We consider three approximation methods: what we call the single cycle approach attributed to Dekker et al and Deshpande et al, the embedded Markov chain approach of Fadigloglu and Bulut, and the continuous time Markov chain approach of Vicil and Jackson.
We confirm the superiority of the embedded Markov chain approach for the case of constant lead times but we find that the fill rates are relatively insensitive to the form of the lead time distribution and both latter approaches, the embedded Markov chain approach and the continuous time Markov chain approach, perform well over wide ranges of lead time variability. For the
three-demand class system, we demonstrate that it is possible to achieve highly differentiated fill rates by demand class and show that these fill rates can be estimated with high accuracy using the continuous time Markov chain approach, provided the fill rate of the lowest priority demand class (the silver fill rate) is not too low.2015-01-08T00:00:00ZTime-changed extremal process as a random sup measureLacaux, CélineSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/379412014-10-10T05:03:36Z2014-10-09T00:00:00ZTitle: Time-changed extremal process as a random sup measure
Authors: Lacaux, Céline; Samorodnitsky, Gennady
Abstract: A functional limit theorem for the partial maxima of a long memory
stable sequence produces a limiting process that can be described as a
beta-power time change in the classical Fr\'echet
extremal process, for beta in a subinterval of the unit
interval. Any such power time change in the extremal process
for 0<beta<1 produces a process with stationary
max-increments. This deceptively simple time change hides the much
more delicate structure of the resulting process as a self-affine
random sup measure. We uncover this structure and show that in a
certain range of the parameters this random measure arises as a limit
of the partial maxima of the same long memory stable sequence, but in
a different space. These results open a way to construct a whole new
class of self-similar Fr\'echet processes with stationary
max-increments.2014-10-09T00:00:00ZTauberian Theory for Multivariate Regularly Varying Distributions with Application to Preferential Attachment NetworksResnick, SidneySamorodnitsky, Gennadyhttp://hdl.handle.net/1813/367122014-06-26T05:04:32Z2014-06-25T00:00:00ZTitle: 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.2014-06-25T00:00:00ZNonstandard regular variation of the in-degree and the out-degree in the preferential attachement modelSamorodnitsky, GennadyResnick, SidneyTowsley, DonDavis, RichardWillis, AmyWan, Phyllishttp://hdl.handle.net/1813/367112014-06-26T05:04:31Z2014-06-25T00:00:00ZTitle: 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.2014-06-25T00:00:00ZGeneral inverse problems for regular variationDamek, EwaMikosch, ThomasRosinski, JanSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/344112013-10-03T05:03:23Z2013-10-02T00:00:00ZTitle: 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.2013-10-02T00:00:00ZStock Optimization in Emergency Resupply Networks under Stuttering Poisson DemandChen, JJackson, P.L.Muckstadt, Jhttp://hdl.handle.net/1813/331862013-04-23T05:03:17Z2013-04-01T00:00:00ZTitle: 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.2013-04-01T00:00:00ZCalculation of ruin probabilities for a dense class of heavy tailed distributionsBladt, MogensNielsen, Bo FriisSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/315402013-03-06T06:01:31Z2013-03-05T00:00:00ZTitle: 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.2013-03-05T00:00:00ZMultivariate tail measure and the estimation of CoVarNguyen, TiloSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/304422012-10-10T05:01:27Z2012-10-09T00:00:00ZTitle: 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.2012-10-09T00:00:00ZFunctional Central Limit Theorem for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative FlowsOwada, TakashiSamorodnitsky, Gennadyhttp://hdl.handle.net/1813/299962012-09-19T05:04:51Z2012-09-18T00:00:00ZTitle: 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.2012-09-18T00:00:00Z