Simulation of Stochastic Processes with Given Accuracy and Reliability

342 Pages·2016·2.1 MB·English

by  Yuriy V. Kozachenko, Oleksandr O. Pogorilyak, Iryna V. Rozora, Antonina M. Tegza

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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ISBN 978-1-78548-217-5

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Contents
Introduct.i..o.n. ........................... . ix
Chapte1r.T heD istribuotfit ohneE stimatfeostr h eN ormo f
Sub-GaussiSatno chasPtriocc esse.s. ..............
1.T1h.se p aocfse u b-Garuasnsdvioaamrn i aanbsdlu ebs- Gaussian
stochparsotcie.cs. s.e.s... ................. 2
.
1.L1E..x ponemnotmieoanfslt u sb -Garuasnsdvioaamrn i ables 8
1.1T.h2se.u o mfi ndepesnudbe-nGtar uasnsdvioaamrn i ables 9
1.31.S. ub-Gasutsoscihpaarnso tcie.cs s.e s. .. . . . . 1.0 . . .
1.T2h.se p aocfse t rsicutbl-yG aruasnsdvioaamrn i aanbsdlt ersi ctly
sub-Gaussian st.o.c.h.a.s..t..i..c. .p.r.o..c.e sse1s5
. .
1.2S.t1r.is cutbl-yG asutsoscihpaarnso tcie.cs .s e.s . . . .2 .2 . . .
13..T hees tiomfca otnevse rrgaetonefscs te r siuctbl-yG aruasnsdioamn
seriinGe( sT). . . . . . . . . . . . . .2 .4 . . . . . . . . . . . . . . . . . .
1AT.h dei streisbtuitomiftao htnnee o sro mfs ub-Gasutsoscihaans tic
proceisLnps (eT.s) .. . . . . . . . . . . . .2 8. . . . . . . . . . . . . . . .
15..T hdei streisbtuitomifato htnnee o sro mfs ub-Gasutsoscihaans tic
proceisnss oemOser lsipcazc. e.s. ................. 30
. .
16..C onverrgaeetnsect ei omfsa ttreisscu tbl-yG aruasnsdsioeamrn i es
inO rlsipcaz.c e.s . . . . . . . . . . . . 3.4 . . . . . . . . . . . . . . . . .
17..S triscutbl-yG aruasnsdsioeamrnw i ietushn corroerol lattheodg onal
ite.m s. . . . . . . . . . . . . . . .4 2. . . . . . . . . . . . . . . . . . .
18..U nifcoornmv eregsetnicomefsa utbe-sG aruasnsdsioeamrn i es 48
19..C onveregsetniocmfesa ttreis cutbl-yG aruasnsdsioeamrn i es
inq T). . . . . . . . . . . . . . . .5 8. . . . . . . . . . . . . .
1.O.1T hees tiomfta htneeo rdmi stroifLb pu-tpiroon.c e.s .s e.s 69
vi SimulatoifoS nt ochasPtriocc essweist Ghi veAnc curacayn dR eliability
Chapte2r. S imulatioofSn t ochastPirco cessePsr esentiendt he
Formo fS erie.s. ............................... ..7 1
2.G1e.n earpaplr ofaocmrho edcseo lnr sutction of
stochparsotciecs ses ................. .7.1. ...........
2.K2a.r hunen-Loetveec henfxiospqriau mneus lioaoftn i on
stochparsotcie.cs. s.e.s. ............... . 73
2.2K.a1r.h unemno-dLoeofsle t vresi uctbl-yG aussian
stochparsotcie.cs. s.e.s. .......... ..... 74
2.2A.c2c.u raanrcdey l iaobfti hKleLi m toydi enLl 2( T.). .. 75
22... 3Accaunrrdae clyi aobfti hKleLim toydi enLl p (Tp )>, 0 75
22..A4c.c uraanrcdey l iaobfti hKleLim toydi enLl u ( T.) 77
22..A5c.c uraanrcdey l iaobfti hKleLim toydi enClC T) 79
2..F 3ourier expafnossrii mounl oatfte icohnn ique
stochparsotcie.cs. s.e.s. ........... ..... 84
2..13F.o urmioedorefs l t riscutbl-yG asutsoscihpaarnso tciecs s 85
2..23A.c curaanrcdey l iaobfti hFlem-i otdyie nLl 2( T.). .. 85
2..33A.c curaanrdce yl iaobfti hFle-i mtoyid nLe pl( Tp )>, 0 86
2..43A.c curaanrcdey l iaobfti hFlem-i otdyie nLl u ( T.) 88
2..53A.c curaanrcdey l iaobfti hFlem-i otdyie nCl ( T) 90
2.S4i.m uloafst tiaotnis otnoacrhpyar sotcwieicst sh
discsrpeetce.t r.u m. . . . . . . . . . . .9 .3 . . . . . . . . . . . . . . . . . .
2.4T.h1me.o doefsl t rsiucbt-lGya sutsastiipaornno acwreiysdt sih s crete
spec.t.r.u.m. .................... ...9.4. ....
2.24A..c curaanrcdey l iaobfti hDle(i Tt)y- mionLd 2e(lT. ). .. 95
2..34A.c curaanrcdey l iaobfti hDle(i Tt)y- moLdpe(lTp )>i, n 0 95
2.4A.c4c.u raanrcdey l iaobfti hDle( i Tt)y- mionLd u(e Tl.) 97
2.4A.c5c.u raanrcdey l iaobfti hDle(i Tt)y- mionCd (eT.l). L10
2.A5p.p liocfFa otiuorenix epra ntsosi iomnu loafst tiaotni onary
stochparsotcie.cs. s.e.s. .............. ....1.0.2
2..15T. hmeo doefals tatipornoacirenwys h sia cc ho rrefluantcitoino n
cabner epreisnteh nfeto erodmfa F oursieerwrii etsh
posictoievfefi c.i.e.n.t.s. .................. . .1.0.3 .
Chapte3r. S imulatioofnG aussiaSnt ochastPirco cessewsi th
RespecttoO utpuPtr ocessoefst heS ystem. ............. 150
31..T hien equfaoltrihe texi peosn meonmteionafttl hs qe u adfroartmisc
ofGa usrsainadnvo amr i.a. b l.e .s . . . . . . . .1 0.7 . . . . . . . . . . . .
32.T. hsep aocfse q uare-rGaanudsvosamir aianan bsdlq eusa re-Gaussian
stochparsotcie.cs. s.e.s. .................. .1.61. .....
3.T3.hd ei stroifsb uuptrieoomnfsu qmusa re-Gaussian
stochparsotcie.cs. s.e.s. ........................ . 117
34..T hees timoafdt iisotnrsfi obsruu tpiroeonmf u m
d
square-sGtaoucshpsarisoatcniei csnt s hees s[0pT,aj .c.e. ...... 1.6 2
Contentsv ii
35.A.c curanardce yl iaobfsi iEmtuylo afGt aiuosnss tioacnh astic
procewsisrteehss petchte topour topcouefsts o sms ey s.t e.m . . 133
36..M odceoln stroufsc ttaitoiGnoa nuasrssyti oacnh parsotcwieicst sh
discsrpeetcewt irrtuehms pteoocu ttp pruotc ess 144
3..S7 imuloafGt aiuosnss tioacnh fiaesl.tdisc 175
.
3..l7S.i mulaGtaiuosnfs iioeaoflnn sd psh eres 116
Chapte4r.T heC onstructoifot nh eM odelo fG aussiaSnt ationary
Processes 169
. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .... . . .
Chapte5r.T heM odelionfgG aussiaSnt ationRaarnyd om
Processweist ah CertaAicnc uracayn dR eliabi.l.i.t.y. 118
5.R1e.E abainladic tcyui rnLa pc(yT p) �, 1 o ft hmeo defloGsra ussian
statiroannadprorymo ces.ses. . . 118
. . . . . . . . . . . . . . . . . . . . . .
5.1T.h1ae.c cuorfma ocdye sltiantgi Goanuasrpsyri oacne sses
inL p(T[]O1), ::, :;P :::;2 . . . . . . . . . . .. .. 1 .8 2. . . . . . . . . . . . .
5. l..T 2haec cuorfma ocdye sltiantgi Goanuasprsryio acne sses
Lp(T[]Oa),pt � 1 . . . . . . . . . . . . . 1.88 . . . . . . . . . . .
5.31.T. haec cuorfma ocdye Glaiunsgss tiaatnir oannadprorymo cesses
inno romfOs r lsipcazc. e.s. .................1.99. ....
5.T2h.e acacnurdre alciyao bfti hmEeot dyse tla tiroannadprorymos csees
int huen ifmoertm.r i.c . . . . . . . . . .. .. 2.0 2. . . . . . . . . . . . . . .
5.2T.h1ae.c cuorfsa icmyu loafst tiaotniG oanuasprsryio acnew sistehs
bounsdpeedc t.r.u.m. ........ ...........2 02
. . . . . . . .
5.2A.p2p.l iocLfap t(iD.op)nr octeshseieonssr i ym uloaGfta iuosns ian
-
statiroannadprorymo ce.s s.e s. . . . . . . .. .. 2 3.1 . . . . . . . . . . .
.
53.A.p pliocfSa Utbicospnp( aD.t)ch ee tofiorn ytd h aec cuorfma ocdye ling
fosrt atiGoanuasprsryi oacne .s s.e s. . . . . . . .2 .2 .2 .
.
54.G. eneramloidozefGel ad u ssstiaatni pornoacreys ses 214
Chapte6r.S imulatoifoCn o xR andomP rocesse.s 215
.
6.R1a.n dCoompx r oce.s .s e.s . . . . . . . . .2 5.1 . . . .
6.S2i.m uloaflto iGgoa nu sCsoipxar nos cseaesasd emaanrdr pirvoacle ss
inac tuarial .m a.t h.e .m a.t i.c s. . . . . .2 .5 3. . . . . . . . . . . . .
.
63..S implmiefitehodofs d i mullaoGtgai unsgCs oipxar no ces.s es2 68
. .
6.4S.i muloaftt ihCoeon x p rocwehsesdn e nsiisgt eyn erbayat ed
homogelnoeGgoa uuss fiseila.dn . . . . . . . . .2 0.8 . . . . . . . . . .
.
65.S.i muloaflto iGgoa nu sCsoipxar no cwehsetsnh d ee nsigiset nye rated
byt hien homogene.o u.s . fi.e l.d . . . . .. .. 2.68 . . . . . . . . . . . .
.
66..S imuloaftt hiCeoo npx r ocwehsetsnh d ee nsiigste yn erbaytt heed
squGaarues rsainadnpo rmo ce.s .s . . . .. .. . .. .. 2.9 .2 . .
.
6..7S imuloaftt ihsoeqn u aGraeu ssCioaxpn r ocwehsesdn e nsiist y
generbayaht oemdo gefineelod.u .s. ... . . . .. .. 2.99 . . .
.
viiiS imulatoifoS nt ochasPtriocc essweist Ghi veAnc curacayn dR eliability
68..S imuloaftt hiseoq nu Gaarues Csoipxar no cwehsetsnh dee nsiist y
generbayat nie ndh omogfieenl.ed o..u s. . . . . .. .. 3.0 1. . . . . . . . .
Chapte7r.O n theM odelionfgG aussiaSnt ationParroyc essweist h
AbsolutCeolnyt inuous Spec.t.ru.m. ...............3.50. .
Chapte8r.S imulatioofGn a ussiaIns otroRpaincd omF ieldosn a
Sphere ... .............................. 351
. . . .
81.S.i muloafrt ainodfinoe mlw di gtihv aecnc uarnardce yl iability
inL2 (S.I l.) . . . . . . . . . . . . . .3 .2 3. . . . . . . . . . . . . . . . . .
8.2S.i muloafrt ainodfinoe mwl idgt ihv aecnc uarnardce yl iability
inL p(SPn ;:)::2, 3 2 4
Bibliography 352
Index 333
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