Simulation of Stochastic Processes with Given Accuracy and Reliability

337 Pages·2016·3.297 MB·English

by  Yuriy V. Kozachenko, Oleksandr O. Pogorilyak, Iryna V. Rozora and Antonina M. Tegza (Auth.)

Simulation of Stochastic Processes with Given
Accuracy and Reliability
Series Editor
Nikolaos Limnios
Simulation of Stochastic
Processes with Given
Accuracy and Reliability

Yuriy Kozachenko
Oleksandr Pogorilyak
Iryna Rozora
Antonina Tegza
First published 2016 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd
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Introduction
The problem of simulation of stochastic process has been a matter of active
research in recent decades. It has become an integral part of research, development
and practical application across many fields of study. That is why one of the actual
problems is to build a mathematical model of stochastic process and study its
properties. Because of the powerful possibilities of computer techniques, the
problemsofnumericalsimulationshavebecomeespeciallyimportantandallowusto
predictthebehaviorofarandomprocess.
There are various simulation methods of stochastic processes and fields. Some
of them can be found in [OGO96, ERM82, CRE93, KOZ07a]. Note that in most
publicationsdealingwithsimulationofstochasticprocesses,thequestionofaccuracy
andreliabilityisnotstudied.
In this book, the methods of simulation of stochastic processes and fields with
given accuracy and reliability are considered. Namely, models are found that
approximate stochastic processes and fields in different functional spaces. This
meansthatatfirstweconstructthemodelandthenusesomeadequacyteststoverify
it.
In most books and papers that are devoted to the simulation of stochastic
processes, the modeling methods of exactly Gaussian processes and fields are
studied.Itisknownthatthereisaneedtosimulatetheprocessesthatareequaltothe
sum of various random factors, in which effects of each other are independent.
According to the central limit theorem, such processes are close to Gaussian ones.
Hence,theproblemofsimulationofGaussianstochasticprocessesandfieldsisahot
topicinsimulationtheory.
Let us mention that in this book only centered random processes and fields are
considered, since simulation of determinate function can be made without any
difficulties.
x SimulationofStochasticProcesseswithGivenAccuracyandReliability
NotethatallresultsinthisbookareapplicableforGaussianprocess.
Chapter1dealswiththespaceofsub-Gaussianrandomvariablesandsubclasses
of this space containing strictly sub-Gaussian random variables. Different
characteristics of these random variables are considered: sub-Gaussian standard,
functionalmoments,etc.Specialattentionisdevotedtoinequalitiesestimating“tails”
of the distribution of a random variable, or a sum of a random variable in the some
functional spaces. These assertions are applied in investigation of accuracy and
reliabilityofthemodelofGaussianstochasticprocess.
In Chapter 2, general approaches for model construction of stochastic processes
with given accuracy and reliability are studied. Special attention is paid to
Karhunen–Loève and Fourier expansions of stochastic processes and their
applicationtothesimulationofstochasticprocesses.
Chapter 3 is devoted to the model construction of Gaussian processes, that is
considered as input processes on some system of filter, with respect to output
processes in a Banach space C(T) with given accuracy and reliability. For this
purpose,square-Gaussianrandomprocessesareconsidered;theconceptofthespace
of square-Gaussian random variables is introduced and the estimates of distribution
of a square-Gaussian process supremum are found. We also consider the particular
casewhenthesystemoutputprocessisaderivativeoftheinitialprocess.
Chapter 4 offers two approaches to construct the models of Gaussian stationary
stochasticprocesses.Themethodsofmodelconstructionaregeneralizedonthecase
ofrandomfields.Theproposedmethodsofmodelingcanbeappliedindifferentareas
ofscienceandtechnology,particularlyinradio,physicsandmeteorology.Themodels
canbeinterpretedasasetofsignalswithlimitedenergy,harmonicsignalsandsignals
withlimiteddurations.
InChapter5,thetheoremsonapproximationofamodeltotheGaussianrandom
process in spaces L1([0,T]) and Lp([0,T]), p > 1 with given accuracy and
reliability are proved. The theorems are considered on estimates of the “tails” of
norm distributions of random processes under different conditions in the space
Lp(T), where T is some parametric set, p ≥ 1. These statements are applied to
investigatethepartitionselectionoftheset[0,Λ]suchthatthemodelapproximatesa
Gaussian process with some accuracy and reliability in the space Lp([0,T]) when
p ≥ 1. A theorem on model approximation of random process with Gaussian with
givenaccuracyandreliabilityinOrliczspaceLU(Ω)thatisgeneratedbythefunction
U isalsopresented.
InChapter6,weintroducerandomCoxprocessesanddescribetwoalgorithmsof
their simulation with some given accuracy and reliability. The cases where an
intensity of the random Cox processes are generated by log Gaussian or square
Introduction xi
Gaussian homogeneous and inhomogeneous processes or fields are considered. We
alsodescribetwomethodsofsimulation.Thefirstoneismorecomplicatedtoapply
inpracticebecauseoftechnicaldifficulties.Thesecondoneissomewhatsimplerand
allows us to obtain the model of the Cox process as a model of Poisson random
variables with parameters that depend on the intensity of the Cox process. The
secondmodelhaslessaccuracythanthefirstmodel.
Chapter 7 deals with a model of a Gaussian stationary process with absolutely
continuousspectrumthatsimulates theprocesswith agivenreliability andaccuracy
in L2(0,T). Under certain restrictions on the covariance function of the process,
formulasforcomputingtheparametersofthemodelaredescribed.
Chapter 8 is devoted to simulation of Gaussian isotropic random fields on
spheres. The models of Gaussian isotropic random fields on n-measurable spheres
are constructed that approximate these fields with given accuracy and reliability in
thespaceLp(Sn), p≥2.
1
The Distribution of the Estimates
for the Norm of Sub-Gaussian
Stochastic Processes
This chapter is devotedto the study of the conditions and rate of convergence of
sub-Gaussian random series in some Banach spaces. The results of this chapter are
used in other chapters to construct the models of Gaussian random processes that
approximate them with specified reliability and accuracy in a certain functional
space. Generally, the Gaussian stochastic processes are considered, which can be
represented as a series of independent items. It should be noted that, as will be
shown, these models will not always be Gaussian random processes. In Chapter 7,
for example, the Gaussian models of stationary processes are sub-Gaussian
processes.TheaccuracyofsimulationisstudiedinthespacesC(T), Lp(T), p > 0,
andOrliczspaceLU(T),whereT isacompact(usuallysegment)andU issomeC-
function. In addition, these models can be used to construct the models of
sub-Gaussian processes that approximate them with a given reliability and accuracy
in a case when the process can be performed as a sub-Gaussian series with
independentitems.Section1.1providesthenecessaryinformationfromthetheoryof
the sub-Gaussian random variables space. Sub-Gaussian random variables were
introducedforthefirsttimebyKahane[KAH60].BuldyginandKozachenkointheir
publication [BUL87] showed that the space of sub-Gaussian random variables is
Banach space. The properties of this space are studied in the work of Buldygin and
Kozachenko[BUL00].Section1.2dealswithnecessarypropertiesofthetheoryfor
strictlysub-Gaussianrandomvariables.In[BUL00],thistheoryisdescribedinmore
detail. Note that a Gaussian centered random variable is strictly sub-Gaussian.
Therefore,allresultsofthissection,aswellasotherresultsofthisbook,obtainedfor
sub-GaussianrandomvariablesandprocessesarealsotrueforthecenteredGaussian
random variables and processes. In section 1.3, the rate of convergence of
sub-Gaussian random series in the space L2(T) is found. Similar results are
2 SimulationofStochasticProcesseswithGivenAccuracyandReliability
containedin[KOZ99b]and[KOZ07a].Section1.4looksatthedistributionestimate
ofthenormofsub-GaussianrandomprocessesinspaceLp(T).Theseestimatesare
alsoconsideredin[KOZ07a].Formoregeneralspaces,namelythespacesSubϕ(Ω)
suchestimatescanalsobefound[KOZ09].Theseestimatesareusedtofindtherate
ofconvergenceofsub-GaussianfunctionalseriesinthenormofspacesLp(Ω).Note
thatinthecasewherep = 2,theresultsofsection1.3arebetterthansection1.4.In
section1.5,theestimatesofdistributionofthesub-Gaussianrandomprocessesnorm
insomeOrliczspacesarefound;insection1.6,theseestimatesareusedtoobtainthe
rate of convergence of sub-Gaussian random series in the norm of some Orlicz
spaces. Similar estimates are contained in [KOZ 99b, KOZ 07a, KOZ 88, ZEL 88,
RYA90,RYA91,TRI91].
The results on the rate of convergence of sub-Gaussian random series in the
Orliczspacethatwerereceivedinsection1.6,aredetailedinsection1.7fortheseries
with either uncorrelated or independent items. In sections 1.8 and 1.9, the rate of
convergence for sub-Gaussian and strictly sub-Gaussian random series in the space
C(T)isobtained.Similarproblemswerediscussedin[KOZ99b]and[KOZ07a].
Section 1.10 provides the distribution estimates for supremum of random
processesinthespaceLp(Ω).
1.1. The space of sub-Gaussian random variables and sub-Gaussian
stochasticprocesses
This section deals with random variables that are subordinated, in some sense,
to Gaussian random variables. These random variables are called sub-Gaussian (the
rigorousdefinitionisgivenbelow).Later,wewillalsostudysub-Gaussianstochastic
processes.
Let{Ω,B,P} beastandardprobabilityspace.
DEFINITION 1.1.– Arandomvariableξ iscalledsub-Gaussian, ifthereexistssuch
numbera≥0thattheinequality
(cid:2) 2 2(cid:3)
a λ
Eexp{λξ}≤exp [1.1]
2
holdstrueforallλ∈R.Theclassofallsub-Gaussianrandomvariablesdefinedona
commonprobabilityspace{Ω,B,P}isdenotedbySub(Ω).
Considerthefollowingnumericalcharacteristicofsub-Gaussianrandomvariable
ξ:
(cid:4) (cid:5)
(cid:2) 2 2(cid:3)
a λ
τ(ξ)=inf a≥0: Eexp{λξ}≤exp ,λ∈R . [1.2]
2
TheDistributionoftheEstimatesfortheNormofSub-GaussianStochasticProcesses 3
Wewillcallτ(ξ)sub-Gaussianstandardofrandomvariableξ.Weputτ(ξ)=∞
ifthesetofa ≥ 0satisfying[1.1]isempty.Bydefinition,ξ ∈ Sub(Ω)ifandonlyif
τ(ξ)<∞.Thefollowinglemmaisclear.
LEMMA1.1.– Therelationshipshold
(cid:6) (cid:7)
1
2lnEexp{λξ} 2
τ(ξ)= sup . [1.3]
λ(cid:2)=0,λ∈R λ2
Forallλ∈R
(cid:4) (cid:5)
2 2
λ τ (ξ)
Eexp{λξ}≤exp . [1.4]
2
Thesub-Gaussianassumptionimpliesthattherandomvariablehasmeanzeroand
imposesotherrestrictionsonmomentsoftherandomvariable.
LEMMA1.2.– Supposethatξ ∈Sub(Ω).Then
E|ξ|p <∞
foranyp>0.Moreover,Eξ =0and
Eξ2 ≤τ2(ξ).
PROOF.– Since as p > 0 and x > 0 the relationship xp ≤ exp{x}ppexp{−p} is
satisfied.Hence,ifinsteadofxwesubstitute|ξ|andtakethemathematicalexpectation
thenobtainthat
E|ξ|p ≤ppexp{−p}Eexp{|ξ|}.
Since
(cid:2) 2 (cid:3)
τ (ξ)
Eexp{|ξ|}≤Eexp{ξ}+Eexp{−ξ}≤2exp <∞,
2
thenE|ξ|p <∞.Further,bytheTaylorformula,weobtain
2
λ
Eexp{λξ}=1+λEξ+ Eξ2+o(λ2),
2
(cid:2) 2 2 (cid:3) 2
λ τ (ξ) λ
2 2
exp =1+ τ (ξ)+o(λ )
2 2
asλ→0.Theninequality[1.4]impliesthatEξ =0andτ2(ξ)≥Eξ2. (cid:2)
4 SimulationofStochasticProcesseswithGivenAccuracyandReliability
Thefollowinglemmagivesanestimateforthemomentsofsub-Gaussianrandom
variable.
LEMMA1.3.– Letξ ∈Sub(Ω),then
(cid:8) (cid:9)
p/2
p
E|ξ|p ≤2 (τ(ξ))p
e
foranyp>0.
PROOF.– Sinceforp>0,x>0theinequality
xp ≤exp{x}ppexp{−p},
holds,thenwecansubstituteλ|ξ|, λ>0forxandtakethemathematicalexpectation
ofsuchavalue.Hence,
(cid:8) (cid:9)
p
p
E|ξ|p ≤ Eexp{λ|ξ|}. [1.5]
λe
Since
Eexp{λ|ξ|}≤Eexp{λξ}+Eexp{−λξ},
thenitfollowsfrom[1.5]and[1.4]thatforanyλ>0theinequality
(cid:8) (cid:9)
p 2 2
p λ τ (ξ)
E|ξ|p ≤2 exp{ }
λe 2
is satisfied. The lemma will be completely proved if in the inequality above we
√
substituteλ= p underwhichtheright-handsideoftheequalityismaximized. (cid:2)
τ(ξ)
EXAMPLE 1.1.– Supposethatξ isanN(0,σ2)-distributedrandomvariable,thatisξ
hasGaussiandistributionwithmeanzeroandvarianceσ2.Then
(cid:4) (cid:5)
2 2
σ λ
Eexp{λξ}=exp ,
2
meaningthatξissub-Gaussianandτ(ξ)=σ.
Example 1.1 and lemma 1.1 show that a random variable is sub-Gaussian if and
only if its moment generating function is majorized by the moment generating
function of a zero-mean Gaussian random variable.This fact somewhat explains the
term “sub-Gaussian”. Note that a function Eexp{λξ} is called moment generating
functionofξ.