研究生水平的测度理论与概率论教材

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"《测度论与概率论》是一本由Krishna B. Athreya和Soumendra N. Lahiri编著的研究生水平教材，涵盖了测度论和概率论的核心内容。这本书适合用于两学期的课程教学，可以选择加入随机过程和其他专题的补充材料。主要目标读者是数学和统计学的博士一年级学生，同时也适用于具有数学背景的工程和经济学学生。必要的预备知识仅限于基本的实变分析概念，如极限、连续性、可微性、黎曼积分以及序列和级数的收敛，这些在附录中都有回顾。" 《测度论与概率论》是SpringerTexts in Statistics系列的一部分，该系列旨在为生命科学和社会科学提供统计学基础。书中的内容深入探讨了测度论的基本原理，包括如何定义和构造测度，以及如何从测度理论出发建立概率空间。概率论部分则涵盖了随机变量、概率分布、期望值、方差、条件概率、独立性、大数定律和中心极限定理等核心概念。书中可能还涉及随机过程，这是概率论的一个重要分支，涵盖了布朗运动、马尔科夫链、泊松过程等经典模型。这些理论在现代科学、工程、金融等领域有广泛应用。此外，作者可能还讨论了统计推断中的关键问题，如估计理论和假设检验，以及随机变量的联合分布和相关性。例如，书中的内容可能涵盖： 1. 测度的构造：从σ代数和勒贝格测度开始，讨论如何在更一般的空间上定义测度。 2. 概率空间：如何构建概率测度，以及如何利用测度论来定义概率事件。 3. 随机变量：定义离散和连续随机变量，讨论它们的分布函数和特征函数。 4. 大数定律和中心极限定理：这些是概率论中的基本定理，对于理解序列的收敛性和随机现象的稳定性至关重要。 5. 马尔科夫过程和布朗运动：描述系统状态随时间演变的随机过程，广泛应用于物理、化学和金融建模。 6. 泊松过程：处理事件发生时间的随机性，常见于排队理论和保险精算。 7. 统计推断：包括参数估计（点估计和区间估计）和假设检验，这些都是数据分析的基础。此外，由于这本书是为研究生设计的，因此可能会包含一些高级主题，如弱收敛、鞅理论、Girsanov变换、随机积分等，这些都是研究和应用中非常重要的工具。《测度论与概率论》是一本深入而全面的教材，不仅介绍了测度论和概率论的基本概念，还提供了将这些理论应用于实际问题的途径，对数学、统计学以及其他相关领域的学生和研究人员来说都是宝贵的资源。

Contents xvii

14.3.3 TheGibbssampler................... 480

14.4Problems ............................ 481

15 Stochastic Processes 487

15.1ContinuoustimeMarkovchains................ 487

15.1.1 Deﬁnition........................ 487

15.1.2 Kolmogorov’s diﬀerential equations .......... 488

15.1.3 Examples ........................ 489

15.1.4 Limittheorems..................... 491

15.2Brownianmotion........................ 493

15.2.1 ConstructionofSBM.................. 493

15.2.2 BasicpropertiesofSBM................ 495

15.2.3 Some related processes ................. 498

15.2.4 Somelimittheorems.................. 498

15.2.5 SomesamplepathpropertiesofSBM ........ 499

15.2.6 Brownianmotionandmartingales .......... 501

15.2.7 Someapplications ................... 502

15.2.8 The Black-Scholes formula for stock price option . . 503

15.3Problems ............................ 504

16 Limit Theorems for Dependent Processes 509

16.1Acentrallimittheoremformartingales ........... 509

16.2 Mixing sequences ........................ 513

16.2.1 Mixing coeﬃcients ................... 514

16.2.2 Couplingandcovarianceinequalities......... 516

16.3 Central limit theorems for mixing sequences ......... 519

16.4Problems ............................ 529

17 The Bootstrap 533

17.1Thebootstrapmethodforindependentvariables ...... 533

17.1.1 A description of the bootstrap method ........ 533

17.1.2 Validityofthebootstrap:Samplemean ....... 535

17.1.3 Secondordercorrectnessofthebootstrap ...... 536

17.1.4 Bootstrapforlatticedistributions .......... 537

17.1.5 Bootstrapforheavytailedrandomvariables..... 540

17.2 Inadequacy of resampling single values under dependence . 545

17.3Blockbootstrap......................... 547

17.4PropertiesoftheMBB..................... 548

17.4.1 ConsistencyofMBBvarianceestimators....... 549

17.4.2 ConsistencyofMBBcdfestimators.......... 552

17.4.3 SecondorderpropertiesoftheMBB ......... 554

17.5Problems ............................ 556

18 Branching Processes 561

18.1 Bienyeme-Galton-Watson branching process ......... 562

2 Measures and Integration: An Informal Introduction

possible. Suppose now that one has the tools to measure the snow exactly

on a class of subsets, such as triangles, rectangles, circular shapes, elliptic

shapes, etc., no matter how small. It is natural to try to approximate oddly-

shaped regions by combinations of these “standard shapes,” and then use

a limiting process to obtain a measure for the oddly-shaped regions and

reach some limit for such sets. Let B denote the class of subsets of S whose

measure is obtained this way and let λ(B) denote the amount of snow in

each B ∈B.CallB the class of all (snow) measurable sets and λ(B)the

measure (of snow) on B for each B ∈B. It is reasonable to expect that the

following properties of B and λ(·)hold:

Properties of B

(i) A ∈B⇒A

∈B(i.e., if one can measure the amount of snow on

A and knows the total amount on S, then one knows the amount of

snow on A

(ii) A

∈B⇒A

∪ A

∈B(i.e., if one can measure the amount of

snow on A

and A

, then one can do the same for A

∪ A

(iii) If {A

: n ≥ 1}⊂B,andA

⊂ A

n+1

for all n ≥ 1, then lim

n→∞

≡



∞

n=1

∈B(i.e., if one can measure the amount of snow on A

for

each n ≥ 1onanincreasing sequence of sets, then one can do so on

the limit of A

(iv) C⊂Bwhere C is the class of nice sets such as triangles, squares, etc.,

that one started with.

Properties of λ(·)

(i) λ(A) ≥ 0forA ∈B(i.e., the amount of snow on any set is nonnega-

tive!)

(ii) If A

∈ B,A

∩ A

= ∅,λ(A

∪ A

)=λ(A

)+λ(A

) (i.e., the

amounts of snow on two disjoint sets simply add up! This property

of λ is referred to as ﬁnite additivity).

(iii) If {A

: n ≥ 1}⊂B,aresuchthatA

⊂ A

n+1

for all n,then

λ(lim

n→∞

)=λ(



∞

n=1

) = lim

n→∞

λ(A

) (i.e., if we can ap-

proximate a set A by an increase sequence of sets {A

}

n≥1

from

B,thenλ(A) = lim

n→∞

λ(A

). This property of λ is referred to as

monotone continuity from below,orm.c.f.b. in short).

This last assumption (iii) is what guarantees that diﬀerent approxima-

tions lead to consistent limits. Thus, if there are two increasing sequences



}

n≥1

and {A



}

n≥1

having the same limit A but {λ(A



)}

n≥1

and

{λ(A



)}

n≥1

have diﬀerent limits, then the approximating procedures are

not consistent.

Measures and Integration: An Informal Introduction 3

It turns out that the above set of reasonable and natural assumptions

lead to a very rich and powerful theory that is widely applicable.

Atriplet(S, B,λ) that satisﬁes the above two sets of assumptions is

called a measure space. The assumptions on B and λ are equivalent to the

following:

On B

B(i)



: ∅, the empty set, lies in B

B(ii)



: A ∈B⇒A

∈B(sameas(i)before)

B(iii)



: A

,...∈B⇒∪

∈B(combines (ii) and (iii) above) (Closure

under countable unions).

On λ

λ(i)



: λ(·) ≥ 0 (same as (i) before) and λ(∅)=0.

λ(ii)



: λ(∪

n≥1



∞

n=1

λ(A

)if{A

}

n≥1

⊂Bare pairwise disjoint,

i.e., A

∩ A

= ∅ for i = j (Countable additivity).

Any collection B of subsets of S that satisﬁes B(i)



, B(ii)



, B(iii)



above

is called a σ-algebra. Any set function λ on a σ-algebra B that satisﬁes

λ(i)



and λ(ii)



aboveiscalledameasure.Thus,ameasure space is a triplet

(S, B,λ)whereS is a nonempty set, B is a σ-algebra of subsets of S and λ is

a measure on B. Notice that the σ-algebra structure on B and the countable

additivity of λ are necessary consequences of the very natural assumptions

(i), (ii), and (iii) on B and λ deﬁned at the beginning.

It is not often the case that one is given B and λ explicitly. Typically,

one starts with a small collection C of subsets of S that have properties

resembling intervals or rectangles and a set function λ on C. Then, B is the

smallest σ-algebra containing C obtained from C by various operations such

as countable unions, intersections, and their limits. The key properties on

C that one needs are:

(i) A, B ∈C⇒A ∩ B ∈C(e.g., intersection of intervals is an interval).

(ii) A ∈C⇒A

is a ﬁnite union of sets from C (e.g., the complement of

an interval is the union of two intervals or an interval itself).

A collection C satisfying (i) and (ii) is called a semialgebra. The function

λ on B is an extension of λ on C. For this extension to be a measure on B,

the conditions needed are

(i) λ(A) ≥ 0 for all A ∈C

(ii) If A

,... ∈Care pairwise disjoint and A =



n≥1

∈C,then

λ(A)=



∞

n=1

λ(A

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