stochastic lectures

7/29/2019 stochastic lectures

1/8

AdvancedStochasticProcesses.DavidGamarnikLECTURE2

Randomvariablesandmeasurable functions. StrongLawof

Large

Numbers

(SLLN).

Scary

stuff

continued

...

OutlineofLectureRandomvariablesandmeasurablefunctions.ExtensionTheorem. Borel-CantelliLemmaandSLLN

1.1. Randomvariablesandmeasurable functionsDefinition 1.1. Given two pairs (1, F1), (2, F2) of a sample space and a -field, a functionX : 1 2 is defined to be measurable if for every A F2 we must have X1(A) F1.When2isthesetofallrealsRandF2 istheBorel-field,thefunctionX iscalledarandomvariable.Thisdefinitionnaturallyextendstothecasewhen2 = d. In thiscasewecall X arandomvector. AlsosincethesetofintegersisasubsetofR,thedefinitionofarandomvariableincludesthecaseofintegervaluedrandomvariables.Exercise1. Supposea functionX :R issuchthatX1(, x) F foreveryrealvaluex. ProvethatX isameasurablefunction.

Note,thatwedonothavetohaveaprobabilitymeasurePon1 or2 inordertodefinemeasurable functions. Butprobabilitymeasure isneededwhenwediscussprobabilitydistributionsbelow.Examples.

(a) Itiseasytogiveanexampleofafunctionwhichisnotmeasurable. Suppose,forexample1 = 2 and both consist of exactly 3 elements 1, 2, 3. Say F1 is a trivial -field(whichconsistsofonly and)andF2 isafull-fieldconsistingofall8subsetsof.Then the identical transformation X : is not measurable: take any non-empty

1


2/8

2 D.GAMARNIK,15.070setA , A=. WehaveAismeasurablewithrespecttoF2,butX1(A)=AisnotmeasurablewithrespecttoF1.

(b) (Figure.) Say=[0, 1]2 andX : RisdefinedbyX()=1+2. WeclaimthatX is a random variable when is equipped with Borel -field. Here is the proof. ForeveryrealvaluexconsiderthesetA={=(1, 2):1+2 >x}. WewillprovethatAismeasurable(belongstotheBorel-fieldof[0, 1]2). ThenwewilltakeacomplementofAandthiswillprovethatX israndomvariable. Considerthecountablesetofpairsofrationals(r1, r2)suchthatr1+r2 >x. Foreachofthemfindn=n(r1, r2)thesmallestintegerwhichislargeenoughsothattherecangle

1 1B(r1, r2;1/n)={(1, 2) [0, 1]2 : 1 r1 , 2 r2| |

n | | n}liesentirely in A (this is possiblebystrict inequalityr1 +r2 >x). Observe that everypair(1, 2)satisfying1 +2 >x lies inoneoftheserectangles. ThusA istheunion

r1,r2B(r1, r2,n(r11,r2))ofthecountablecollectionofsuchrectanglesandthereforebelongstotheBorel-fieldof[0, 1]2. (c) Say = C[0, ) equipped with the Borel -field, and X : R is the functionwhich maps every continuous function f(t) into max0t1 f(t) . Then X is a random| |variable on . Indeed, for every x, we have X1(x) is the set of all functions f suchthatmax0t1 f(t) . Butthis isexactlythesetB(0, x, 1)used inDefinition1.5of x| |Lecture1. ThesetsofthistypegeneratetheBorel-field,and inparticular,belongtoit. ThusX ismeasurable.

TheconceptofrandomvariablesnaturallyleadstotheconceptofprobabilitydistributionDefinition1.2. (Figure.) Givenaprobabilityspace(, F,P)andarandomvariablesX :R, the associatedprobabilitydistribution is defined to be the function F :R [0, 1] given byF(x) =P({ : X() x}). When F(x) is a differentiable function of x, its derivativef(x)=F (x)iscalledthedensityfunction.InotherwordsF(x) istheprobabilitygiventothesetofallelementaryoutcomes whicharemappedbyX intovalueatmostx.It istheprobabilitydistributionswhichareusuallydiscussed inelementaryprobabilityclasses.There,oneusuallydefinesprobabilitydistributionasafunctionsatisfyingcertainproperties(likeitshouldbenon-decreasingandshouldconvergetounityasx ). Herethesepropertiescanbederivedfromthegivendefinitionofaprobabilitydistribution.Proposition1. ProvethatF(x)isnon-decreasing,non-negativeand

F(x)=0, limxF(x)=1.limxProof. HW

Theconceptofprobabilitydistributionsallowsonetoperformtheprobabilityrelatedcalculationswithoutalludingtomoreabstractnotionsofprobabilitymeasures. Thisisnotpossible,however,whenwediscussprobabilityspaceslikeC[0, ).


3/8

LECTURE2.PROBABILITYBASICSCONTINUED 3Havingdefinedrandomvariablesandassociatedprobabilitydistributions,wecandefinefurtherexpectedvalues,moments,momentgeneratingfunctions,etc.,inamoreformalwaythenisdoneinelementaryprobabilityclasses. Wedothisonlyheuristically,highlightingthemainideas.Definition 1.3. A random variable X : R is called simple if it takes only finitely manyvalues

x1, x2, . . . , xm.

The

expected

value

of

asimple

random

variable

X

is

defined

to

be

the

quantity

E[X] = xiP{ :X()=xi}.1im

What if X is not simple? How do we define its expected value? The idea is to approximateX by a sequence of simple random variables. For simplicity assume that X takes only valuesin the interval [0, A] for some A > 0. That is X : [0, A]. Now consider Xn() = k ifn

kX() (k1, ]. Then Xn is a simple random variable. It can be shown that the sequence ofn nthe correspondingexpectedvaluesE[Xn]converges. Its limit iscalled theexpectedvalueE[X]ofX. Itisalsosometimeswrittenas

X()dP(). Thisdefinitionofexpectedvaluesatisfiesall

thepropertiesofexpectedvaluesonestudiesinelementaryprobabilitycourses,forexamplethefactE[X2] (E[X])2,Markovinequality,Chebyshevinequality,Jensensinequality,etc.1.2. Whats i.i.d. sequenceofrandomvariables?Nowwecangiveaformaldefinitionofastochasticprocess theprinciplenotionforthiscourse.Definition 1.4. Let T be the set of all non-negative realsR+ or integersZ+. A stochasticprocess{Xt}tT isafamilyofrandomvariablesXt : RparametrizedbyT.Remark. Note that a sample outcome corresponding to a stochastic process is a functionX() : T R, and the sample space of the corresponding stochastic process is the space offunctions fromT intoR. Butoftenweconsiderrestrictions. Forexample,whenT =[0, )wemightconsideronlycontinuousfunctionsfrom[0, )intoR:Example. Set=C[0, )equippedwithBorel-field. DefineXt()=(t)foreverysample C[0, ). Then {X t[0,) is a stochastic process. This is true because each functiont}Xt :C[0, Risarandomvariable. (wewillprovethislaterinthecourse).)Remark. ThedefinitionnaturallyextendstothecasewhenobservationsarefunctionsT Rdintod-dimensionalEuclidianspace.One of the simplest(to analyze, but not to define) examples of a stochastic process is an i.i.d.(independent,identicallydistributed)stochasticprocess. Whatisani.i.d. stochasticprocess? Inprobabilitycourses itwascommontosayX1, X2, . . . , isan i.i.d. sequenceofBernoullirandomvariables with parameter 0


4/8

4 D.GAMARNIK,15.070somecollectionofsetsA andwecandefinePonA only,thethereisauniqueextensionofthefunctionPontoentire-field,providedsomerestrictionsaresatisfied.1.2.1. ExtensionTheoremTheorem1.5 (ExtensionTheorem).GivenasamplespaceandacollectionA ofsubsetsofsuch thatforeveryA A itscomplement \ A isalso in andforeveryfinitesequenceA1, . . . , Am itsunion1jmAj isalso inA. SupposeP:A [0, 1] issuch that

(a)P()=1,(b)P(j=1Aj) P(Aj),whenever .j=1 j=1Aj A(c)P( Aj)= P(Aj),whenever Aj A andAi, i=1, 2, . . .aremutuallyexclu-j=1 j=1 j=1

sive.Then thefunctionPuniquelyextends toaprobabilitymeasureP:F(A) [0, 1]definedon the-fieldgeneratedbyA.Remark. Note, that the requirement from A is to be a collection of sets with properties verysimilartothatofa-field. Theonlydifferenceisthatwedonotrequireevery infiniteunionofsetstobeinA aswell.1.2.2. ExamplesandapplicationsUniformprobabilitymeasure. Consider = [0, 1] and let A be the set of finite unions ofopenorclosednon-intersectingintervals: [a1, b1) [a2, b2] m, bm). Itiseasytocheckthat (aA satisfiestheconditionsoftheET.ConsiderthefunctionP:A [0, 1]whichmapseverysuchset of intervals to the value1imbi ai (that is the total length of these intervals). It canbecheckedthatthis alsosatisfiestheconditionsoftheET(weskiptheproof). Thus,byET,there exists a unique extension of functionP to a probability measure on entire Borel -field,sincethis-fieldisgeneratedbyintervals. Thisprobabilitymeasureiscalleduniformprobabilitymeasureon [0, 1].Other types of continuous distributions. What about other distributions like Normal,Exponential, etc.? The proper definition of these probability measures is introduced similarly.For example the standard Normal distribution is defined as probability space (R, B,P), where

21 t

2B is the Borel -field onR andP assigns to each interval [a, b] valuea

b2e dt. Then each

non-intersectingcollectionofintervals[ai, bi], 1 i misassignedvaluewhichisthesumofthecorresponding integrals. Again the setoffinite collectionsofnon-intersecting intervals satisfiesthe conditions ofET, andapplyingETweobtainedthatthe probabilitymeasureP isdefinedon

the

entire

Borel

-field

B.

1.2.3. i.i.d. sequencesi.i.d coin tosses . Let ={0, 1}. Recall that the product -field is the field generated bycylindertypesetsA(). LetA bethesetoffiniteunionsofsuchsets1jkA(j). Again,itcanbecheckedthatthatA satisfiestheconditionsofET.Foreveryfinitesequence=1, . . . , m

1andthecorrespondingsetA()wesetP(A())simplytobe2m (theprobabilityofaparticular

1sequence of 0/1 observations in the first m coin tosses is2m). For example, the probability of

first four zeros is214. Then, for every union of non-intersecting sets 1jkA(j) we set their


5/8

5LECTURE2.PROBABILITYBASICSCONTINUEDkcorrespondingvalueto

2m. TheconditionsofETagaincanbechecked,butweskiptheproof.Then,byETthereisauniqueextensionofPtotheentireproduct-fieldof. Thisiswhatwecallasequenceofi.i.d. unbiasedcointosses alsoknownasasequenceofi.i.d.Bernoullirandomvariableswithparameter1/2. Thephrasei.i.d.,inproperprobabilisticterms,means(, F,P)theprobabilityspaceconstructedabove.General i.i.d. typedistributions. Wehavedefinedformallyi.i.d. Bernoullisequence. Whataboutgeneral i.i.d. sequences? Theyare definedsimilarlybyconsidering infiniteproductsandcylindertypesets.First we set = R. On it we consider the product -field F. Define A to be the setof finite unions of cylinder type sets. Recall that a cylinder set A is the set of the formA = [a1, b1] [am, bm)R product of closed or open or half-closed half(a2, b2) open intervals. Recall also that cylinder sets generate, by definition, the product -field F.Suppose we have a probability space (R, B,P) defined onR and its Borel -field B (for exampleP corresponds to standard Normal distribution). Then for every cylinder set A we defineP(A)=1jmP([aj, bj]). AgainwecheckthatAandPsatisfytheconditionsofET(weskiptheproof). ThusthereisauniqueextensionofPtotheentireproduct-fieldF ofR,sinceA generates this -field. Then we define Xm() =m for every R. We note that Xm isa random variable as it is a measurable function fromR intoR. The sequence X1, X2, . . . isa stochastic process which we call an i.i.d. sequence of random variables. Essentially we haveembeddedasequenceofrandomvariables{Xm}intoasingleprobabilityspace(R, F,P).Isthisdefinitionconsistentwithelementarydefinitionofi.i.d. Recallthatelementarydefinitionof i.i.d. sequence iswhenP(X1 x1, . . . , X m xm) =1jmP(Xj xj). Isthis true inourcase? Note

P(X1 x1, . . . , X n m)=P{R :1 (, x1], . . . , m m]x (, x }] m]=P{(, x1 (, x R}

= P((, xj]),1jm

wherethelastequalityfollowsfromhowwedefinedPoncylindersets. Buttheproductoftheseprobabilitiesisexactly1jnP(Xj xj). Thustheidentitychecks.

1.3.Borel-Cantelli

Lemma

and

Strong

Law

of

Large

Numbers

(SLLN)

TheStrongLawofLargeNumbers(SLLN)(liketheCentralLimitTheorem)isoneofthemostfundamentaltheorems inprobabilitytheory. Yet properlystating it, letaloneproving it isnotasstraightforwardasis,forexampletheWeakLawofLargeNumbers(WLLN).Wenowusethe(, F,P)frameworktoproperlystateandproveSLLN.Webeginwithaveryusefultool,theBorel-CantelliLemma. Givenasamplespace,a-fieldF and an infinite sequence A1, A2, . . . , Am, . . .F define Ai.o. (i.o. stands for infinitely often)to betheset of all whichbelong to infinitelymany Am-s. One can write Ai.o. as (check


6/8

6 D.GAMARNIK,15.070thevalidityofthisidentity)

Ai.o. = m1 jm Aj.Lemma 1.6 (Borel-Cantelli Lemma). Given a probability space (, F,P) and an infinitesequenceof events Am, m 1 suppose P(Am) )=P((

1in )4 >4)|

n nE[(

1inYi)4]n44

WhenweexpandE[(1inYi)4]wenotethatonlythetermsoftheformE[Y4]andE[Y2Y2]arei i jnon-zero,sincetheexpectedvalueofYi iszeroandthesequence is i.i.d. Alsoby independence


7/8

7LECTURE2.PROBABILITYBASICSCONTINUEDE[Y2Yj2] =(E[Y12])2. Weobtainaboundi

2nE[Y4] +n(n 1)(E[Y12])2 n [E[Y4] +(E[Y12])2] E[Y4] +(E[Y12])21 1 1=n44 n44 n24

This expression is finite by our assumption of finiteness of fourth moment. Since the sumE[Y

1

4]+(E[Y1

2])2n1 n24 n0 wemust have P1inn | . Thismeansthatforalmostevery,wehave |

Yi()

1inlim =0.nn

Thisconcludestheproof. 1.4. Readingassignments

NotesModeling

experiments,

pages

1.4,1.5,2.2.

GrimmettandStirzaker[2],Chapters1and2. Chapter7,Sections7.3-7.5. Durrett[1]Chapter1,Sections1-7.


8/8

BIBLIOGRAPHY1. R.Durrett,Probability: theoryandexamples,DuxburyPress,secondedition,1996.2. G.R.GrimmettandD.R.Stirzaker,Probabilityandrandomprocesses,OxfordSciencePub

lications,1985.

9

stochastic lectures

Documents