《关于鞅的一些主题和模型》郝顺利|(epub+azw3+mobi+pdf)电子书下载

图书名称：《关于鞅的一些主题和模型》

【作　者】郝顺利

【页 数】 137

【出版社】 北京：旅游教育出版社 , 2021.01

【ISBN号】978-7-5637-4204-2

【价 格】60.00

【参考文献】 郝顺利. 关于鞅的一些主题和模型. 北京：旅游教育出版社, 2021.01.

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《关于鞅的一些主题和模型》内容提要：

本书主要研究关于鞅的一些特别的主题和模型。除了第一部分的预备知识外，主要内容分为五部分，分别对Banach空间值鞅差三角阵列的双指标随机加权和在大数定律中的完全收敛，Banach空间值鞅差三角阵列的双指标随机加权和在大数定律中的完全矩收敛，随机环境中分枝测度的局部维数，随机环境中乘积瀑布的极限定理，随机环境中定向聚合物自由能的收敛进行了精细的研究。

《关于鞅的一些主题和模型》内容试读

Chapter 0

Preliminary knowledge

In this chapter,we introduce some basic concepts of probability theory andtheir elementary properties,we also introduce some probabilistic models.We canfind them in some classical books(see,for example,[74,17])and new papers.

0.1Probability spaces and random variables (el-

ements)

Definition 0.1 Let be a nonempty set.Let F be a o-field of subsets of S,thatis,a nonempty class of subsets ofn which contains n and is closed under countableunion and complementation.Let P be a measure defined on F satisfying P(S)=1.

Then the triple (F,P)is called a probability space,and P,a probability measure.

The set is the sure event,and elements of F are called events.Singletonsets {w}are called elementary events.The symbol 0 denotes the empty set andis known as the null or impossible event.Unless otherwise stated,the probabilityspace (F,P)is fixed,and A,B,C,...,with or without subscripts,representevents.

We note that,ifAn∈F,n=l,2,,then A号，Ue1An,∩2An,liminfn An,limsupn An and limno An (if it exists)are events.Also,the probability measure P is defined on F,and for all events A,An,

P(A)≥0，

(但A-2r)(-1

2

CHAPTER 0.PRELIMINARY KNOWLEDGE

It follows that

P(0)=0,P(A)>P(B)for AC B

Moreover,

P(lim inf An)≤liminfP(An)≤limsupP(An)≤P(lim sup An

、7n+00

n+00

n-00

n→00

and,if limn An exists,then

P（lim An)=imP(An).

n+0

The last result is known as the continuity property of probability measures.

Example 0.1 Let {wj j>1},and let F be the o-field of all subsetsof n.Let {pj,j>1}be any sequence of nonnegative real numbers satisfying∑g1n=i.Define P on F by

P(E)=∑P5,E∈F

w∈E

Then P defines a probability measure on (S,F),and (F,P)is a probability space.

Example 0.2 Let =(0,1]and F=B be the o-field of Borel sets on Letbe the Lebesgue measure on B.Then (S,F,A)is a probability space.

Definition 0.2 Let (S,F,P)be a probability space.A real-valued function Xdefined on n is said to be a random variable if

X-1(E)={w∈n:X(w)∈E}∈F for all E∈B

where B is the a-field of Borel sets in R=(-oo,o);that is,a random variable

X is a measurable transformation of (S,F,P)into (R,B1).

We note that it suffices to require that X1(I)EF for all intervals I in R,orfor all semiclosed intervals I =(a,b],or for all intervals I =(-o,b],and so on.

Unless otherwise specified,X,Y,...,with or without subscripts,will representrandom variables.

We note that a random variable X defined on (n,F,P)induces a measure Pxon B defined by the relation

Px(E)=P(X-(E))(E EB1).

Clearly Px is a probability measure on B and is called the probability distributionor,simply,the distribution of X.We note that Px is a Lebesgue-Stieltjes measureon B1.

0.1.PROBABILITY SPACES AND RANDOM VARIABLES (ELEMENTS)3

Definition 0.3 For every x ER set

Fx(x)=Px(-oo,x=P{w∈D:X(w)≤x.

(0.1)

We call Fx =F the distribution function of the random variable X.

The following theorem is an elementary property of a distribution function.

Theorem 0.1 The distribution function F of a random variable X is a nonde-creasing,right-continuous function on R which satisfies

F(-oo)=lim F(x)=0

and

F(+oo)=lim F(x)=1.

Corollary 0.1 A distribution function F is continuous at x E R if and only if

P{w:X(w)=x}=0.

Remark 0.2 Let X be a random variable,and let g be a Borel-measurable func-tion defined on R.Then g(X)is also a random variable whose distribution isdetermined by that of X.

The following theorem show that a function F on R with the properties statedin Theorem 0.1 determines uniquely a probability measure Pr on B1.

Theorem 0.3 Let F be a nondecreasing,right-continuous function defined on Rand satisfying

F(-0o)=0

and

F(+∞)=1.

Then there exists a probability measure P=Pg on By determined uniquely by therelation

PF(-o0,]F(x)

for every x∈R.

Remark 0.4 Let F be a bounded nondecreasing,right-continuous function definedon R satisfying F(-oo)=0.Then there exists a finite measure u=ur on B1determined uniquely by the relation

LF(-o0,]=F(x)

for every x∈R.

CHAPTER 0.PRELIMINARY KNOWLEDGE

Remark 0.5 Let F on R satisfy the conditions of Theorem 0.3.Then there existsa random variable X on some probability space such that F is the distributionfunction of X.In fact,consider the probability space (S,B,P),where P is theprobability measure as constructed in Theorem 0.3.Let X(w)=w,for all wER.

It is easy to see that F is the distribution function of the random variable X

Let F be a distribution function,and let x ER be a discontinuity point of F.

Then p(x)=F(x)-F(x-0)is called the jump of F at x.A point is said to bea point of increase of F if,for every >0,F(x+e)-F(x-e)>0.

Some other elementary properties of a distribution function are obtained in thefollowing propositions.

Proposition 0.1 Let F and F2 be two distribution functions such that

(x)=F2(x)for all x∈D,

where D is everywhere dense in R.Then Fi(x)=F2(x)for every x ER.

Proposition 0.2 The set of discontinuity points of a distribution function F iscountable

Proposition 0.3 Let F1 and F2 be two distribution functions,and let C1 and C2,respectively,be their sets of continuity points.If

F(x)=F2(x)forx∈C∩C2,

then(x)=2(x）for all x∈R.

Let (F,P)be a probability space,and B be a real Banach space with norm

Clearly B is a topological vector space with respect to the metric topologyinduced by l.Let B2 be the o-field generated by the class of all open subsetsof B.Then B2 is known as the Borel o-field on B,and elements of B2 are called

Borel sets.

Definition 0.4 A mapping X:-B is called a B-valued random element if Xis B2-measurable,that is,for every EE B2,

X-1(E)={w∈2：X(w)∈E∈F.

Remark 0.6 Let X:(F,P)(B,B2)be a B-valued random element,and let

Bo be another Banach space with Borel o-field Bo.Let T be a measurable mappingof (B,B2)into (Bo,Bo).Then T(X)is a Bo-valued random element.

Remark 0.7 Let B*be the dual (or conjugate)of B,that is,B*is the Banachspace consisting of all bounded (continuous)linear functional on B.Then it followsimmediately from Remark 0.6 that,for every IE B*,l(X)is a real-valued random.variable.In particular,if B is separable and l(X)is a random variable for everyI E B*,then X is a B-valued random element.

0.2.EXPECTATION AND CONDITIONAL EXPECTATION

5

Proposition 0.4 Let B be a separable Banach space,and X a B-valued randomelement.Then X is a random variable.

Let Px be the set function on B2 defined by

Px(E)=P(X-(E),EEB2.

Clearly Px is a probability measure on B2 induced by X and is known as theprobablity distribution of X.

Definition 0.5 Let X and Y be two B-valued random elements defined on (S,F,P)

Then X and Y are said to be identically distributed if Px Py.A collection of

B-valued random elements is said to be identically distributed if every pair has thesame probability distribution.In particular,X is said to be symmetric if X and

-X are identically distributed.Here-X is the B-valued random element definedby (-X)(w)=-X(w)for allw En.

Remark 0.8 Let {Xa,a E A be a collection of identically distributed B-valuedrandom elements,and let T be a measurable mapping of B-Bo,where Bo isa Banach space.Then {T(Xa),a EA}is a collection of identically distributed

Bo-valued random elements.In particular,if B is separable and,for every l E B*,(Xa),a E A}is a collection of identically distributed random variables,then

Xa,aEA}are identically distributed.

0.2Expectation and conditional expectation

We now study some characteristics of a random variable (or of its distributionfunction).These play an important role in the study of probability theory.

Let (F,P)be a probability space,and X be a random variable defined onit.Let g be a real-valued Borel-measurable function on R.Then g(X)is also arandom variable.

Definition 0.6 We say that the mathematical expectation (or,simply,the expec-tation)of g(X)erists if g(X)is integrable over n with respect to P.In this casewe define the expectation Eg(X)of the random variable g(X)by

0=g(X(w))dP(w)=/g(x)dP.

Remark 0.9 Since integrability is equivalent to absolute integrability,it followsthat Eg(X)exists if and only if Elg(X)erists.

Let Px be the probability distribution of X.Suppose that Eg(X)exists.Thenit follows that g is also integrable over R with respect to Px.Moreover,the relation

(-g(t)dPx(t)

(0.2)

6

CHAPTER 0.PRELIMINARY KNOWLEDGE

holds.We note that the integral on the right hand side of(0.2)is the Lebesgue-

Stieltjes integral of g with respect to Px.

In particular,if g is continuous on R and Eg(X)exists,we can rewrite (0.2)asfollows:

g(x)dp=s

g(z)dF(x).

(0.3)

-0∞

where F is the distribution function corresponding to Px,and the last integral isa Riemann-Stieltjes integral.Two important special cases of (0.3)are as follows.CASE 1.Let F be discrete with the set of discontinuity points {n,n=1,2,...}Let p(xn)be the jump of F at zn,n =1,2,....Then Eg(X)existsif and only if(n)p(n)<,and in that case we have

Eg(X)=∑9(rnp(rn)

n=

CASE 2.Let F be absolutely continuous on R with probability density function

f()=F().Then Eg(x)exists if and only if()f()dz<,and inthat case we have

n00

Eg(X)=

g(x)f(x)dx.

We now state some elementary properties of random variables with finite ex-pectations which follows as immediate consequences of the properties of integrablefunctions.Denote by L1 =L1(,F,P)the set of all random variables X onsuch that EX

(d)LetX∈Ll.Then EX|=0÷X=0a.s.

(e)For EE F,write 1E for the indicator function of the set E,that is,1E =1on E and=0 otherwise..Then X∈Ll→XlE∈Ll,and we write

XdP=E(X1E)

JE

Also,E(X1E）=0÷either P(E)=0orX=0a.s.onE.(f)IfX∈Ll,thenX=0a.s.÷E(X1E)=0 for all E∈F(g)Let X EL,and define the set function Qx on F by

Then Qx is countably additive on F,and Qx is absolutely continuous with respectto P.In particular,Qx is a finite measure on F if X >0 a.s.

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