概率论：理论与示例-第4版-修订4.1版-2013年4月_Durret

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"Probability: Theory and Examples – 4th Ed-Version 4.1-Apr. 21, 2013_Durret" 是一本由Rick Durret编写的概率论教材，该版本修复了截至2013年4月13日前的所有已知印刷错误，并新增了第8章的三个新部分。本书是数学研究生级别的教科书，涵盖了概率论的基础理论和实例。主要知识点包括： 1. **测度论**： - **概率空间**：定义了概率论的基础框架，包括样本空间、事件和概率测度。 - **分布**：讨论了随机现象的概率分布，如均匀分布、正态分布等。 - **随机变量**：引入了随机变量的概念，它是概率空间上的实值函数，用于量化不确定性。 2. **积分理论**： - **Lebesgue积分**：介绍如何对随机变量进行积分，以便计算其期望值和其他统计量。 - **积分的性质**：探讨积分的线性性质、保序性等。 - **积分极限定理**：展示了积分与极限之间的关系。 3. **期望值**： - **不等式**：如Markov不等式、Chebyshev不等式等，它们在估计概率和稳定性分析中非常有用。 - **积分到极限**：讨论如何通过积分来求解极限问题。 - **期望值计算**：提供了计算随机变量期望值的方法，包括离散型和连续型随机变量。 4. **乘积测度与Fubini定理**： - **乘积测度**：描述两个或多个独立随机过程的联合概率。 - **Fubini定理**：允许在两个测度空间之间交换积分，是处理多维概率问题的关键工具。 5. **大数定律**： - **独立性**：介绍了独立随机变量的基本概念和性质。 - **弱大数定律**：阐述了样本均值在多次独立重复实验后趋向于期望值的统计规律。 - **强大数定律**：更强大的版本，保证几乎必然收敛。 6. **随机序列的收敛**： - **随机序列的收敛**：探讨随机变量序列的各类收敛性，如几乎处处收敛、Lp收敛等。 - **收敛速率**：研究收敛的速度，对于理解统计效率至关重要。 - **无穷期望**：处理期望值为无穷的随机变量序列的情况。 7. **Borel-Cantelli引理**： - 这是概率论中的一个重要工具，用于判断一系列事件发生次数的极限行为。 8. **其他主题**（由于内容未完整提供，这部分无法详细展开）：可能包括中心极限定理、大偏差理论、遍历理论等内容，这些都是概率论的重要组成部分，通常涉及随机过程的长期行为和极限定理。这本书深入浅出地讲解了概率论的核心概念，适合对概率论有较高需求的数学和统计学学生以及相关领域的专业人士学习。

10 CHAPTER 1. MEASURE THEORY

−1

(x)

−1

(y)







Figure 1.4: Picture of the inverse deﬁned in the proof of Theorem 1.2.2.

Even though F may not be 1-1 and onto we will call X the inverse of F and denote

it by F

−1

. The scheme in the proof of Theorem 1.2.2 is useful in generating random

variables on a computer. Standard algorithms generate random variables U with a

uniform distribution, then one applies the inverse of the distribution function deﬁned

in Theorem 1.2.2 to get a random variable F

−1

(U) with distribution function F .

If X and Y induce the same distribution µ on (R, R) we say X and Y are equal

in distribution. In view of Theorem 1.1.2, this holds if and only if X and Y have

the same distribution function, i.e., P (X ≤ x) = P (Y ≤ x) for all x. When X and Y

have the same distribution, we like to write

= Y

but this is too tall to use in text, so for typographical reasons we will also use X =

Y .

When the distribution function F (x) = P (X ≤ x) has the form

F (x) =

−∞

f(y) dy (1.2.1)

we say that X has density function f. In remembering formulas, it is often useful

to think of f (x) as being P (X = x) although

P (X = x) = lim

→0

x+

x−

f(y) dy = 0

By popular demand we have ceased our previous practice of writing P (X = x) for the

density function. Instead we will use things like the lovely and informative f

(x).

We can start with f and use (1.2.1) to deﬁne a distribution function F . In order

to end up with a distribution function it is necessary and suﬃcient that f(x) ≥ 0 and

f(x) dx = 1. Three examples that will be imp ortant in what follows are:

Example 1.2.1. Uniform distribution on (0,1). f(x) = 1 for x ∈ (0, 1) and 0

otherwise. Distribution function:

F (x) =











0 x ≤ 0

x 0 ≤ x ≤ 1

1 x > 1

Example 1.2.2. Exponential distribution with rate λ. f(x) = λe

−λx

for x ≥ 0

and 0 otherwise. Distribution function:

F (x) =

(

0 x ≤ 0

1 − e

−x

x ≥ 0

12 CHAPTER 1. MEASURE THEORY

In Section 1.6, we will see the Bernoulli, Poisson, and geometric distributions. The

next example shows that the distribution function associated with a discrete proba-

bility measure can be quite wild.

Example 1.2.6. Dense discontinuities. Let q

, q

, ... be an enumeration of the

rationals. Le t α

> 0 have

∞

i=1

= 1 and let

F (x) =

∞

i=1

,∞)

where 1

[θ,∞)

(x) = 1 if x ∈ [θ, ∞) = 0 otherwise.

Exercises

1.2.1. Suppose X and Y are random variables on (Ω, F, P ) and let A ∈ F . Show

that if we let Z(ω) = X(ω) for ω ∈ A and Z(ω) = Y (ω) for ω ∈ A

, then Z is a

random variable.

1.2.2. Let χ have the standard normal distribution. Use Theorem 1.2.3 to get upper

and lower bounds on P (χ ≥ 4).

1.2.3. Show that a distribution function has at most countably many disc ontinuities.

1.2.4. Show that if F (x) = P (X ≤ x) is continuous then Y = F (X) has a uniform

distribution on (0,1), that is, if y ∈ [0, 1], P (Y ≤ y) = y.

1.2.5. Suppose X has continuous density f, P (α ≤ X ≤ β) = 1 and g is a func-

tion that is strictly increasing and diﬀerentiable on (α, β). Then g(X) has density

f(g

−1

(y))/g

−1

(y)) for y ∈ (g(α), g(β)) and 0 otherwise. When g(x) = ax + b with

a > 0, g

−1

(y) = (y − b)/a so the answer is (1/a)f((y − b)/a).

1.2.6. Suppose X has a normal distribution. Use the previous exercise to compute

the density of exp(X). (The answer is called the lognormal distribution.)

1.2.7. (i) Suppose X has density function f. Compute the distribution function

of X

and then diﬀerentiate to ﬁnd its density function. (ii) Work out the answer

when X has a standard normal distribution to ﬁnd the density of the chi-square

distribution.

1.3 Random Variables

In this section, we will develop some results that will help us later to prove that

quantities we deﬁne are random variables, i.e., they are measurable. Since most of

what we have to say is true for random elements of an arbitrary measurable space

(S, S) and the proofs are the same (sometimes easier), we will develop our results in

that generality. First we need a deﬁnition. A function X : Ω → S is said to be a

measurable map from (Ω, F) to (S, S) if

−1

(B) ≡ {ω : X(ω) ∈ B} ∈ F for all B ∈ S

If (S, S) = (R

, R

) and d > 1 then X is called a random vector. Of course, if

d = 1, X is called a random variable, or r.v. for short.

The next result is useful for proving that maps are measurable.

1.3. RANDOM VARIABLES 13

Theorem 1.3.1. If {ω : X(ω) ∈ A} ∈ F for all A ∈ A and A generates S (i.e., S

is the smallest σ-ﬁeld that contains A), then X is measurable.

Proof. Writing {X ∈ B} as shorthand for {ω : X(ω) ∈ B}, we have

{X ∈ ∪

} = ∪

{X ∈ B

}

{X ∈ B

} = {X ∈ B}

So the class of sets B = {B : {X ∈ B} ∈ F} is a σ-ﬁeld. Since B ⊃ A and A generates

S, B ⊃ S.

It follows from the two equations displayed in the previous proof that if S is a

σ-ﬁeld, then {{X ∈ B} : B ∈ S} is a σ-ﬁeld. It is the smallest σ-ﬁeld on Ω that

makes X a measurable map. It is called the σ-ﬁeld generated by X and denoted

σ(X). For future reference we note that

σ(X) = {{X ∈ B} : B ∈ S} (1.3.1)

Example 1.3.1. If (S, S) = (R, R) then possible choices of A in Theorem 1.3.1 are

{(−∞, x] : x ∈ R} or {(−∞, x) : x ∈ Q} where Q = the rationals.

Example 1.3.2. If (S, S) = (R

, R

), a useful choice of A is

{(a

, b

) × ··· ×(a

, b

) : −∞ < a

< b

< ∞}

or occasionally the larger collection of open sets.

Theorem 1.3.2. If X : (Ω, F) → (S, S) and f : (S, S) → (T, T ) are measurable

maps, then f(X) is a measurable map from (Ω, F) to (T, T )

Proof. Let B ∈ T . {ω : f(X(ω)) ∈ B} = {ω : X(ω) ∈ f

−1

(B)} ∈ F, since by

assumption f

−1

(B) ∈ S.

From Theorem 1.3.2, it follows immediately that if X is a random variable then so

is cX for all c ∈ R, X

, sin(X), etc. The next result shows why we wanted to prove

Theorem 1.3.2 for measurable maps.

Theorem 1.3.3. If X

, . . . X

are random variables and f : (R

, R

) → (R, R) is

measurable, then f(X

, . . . , X

) is a random variable.

Proof. In view of Theorem 1.3.2, it suﬃces to show that (X

, . . . , X

) is a random

vector. To do this, we observe that if A

, . . . , A

are Borel sets then

{(X

, . . . , X

) ∈ A

× ··· × A

} = ∩

∈ A

} ∈ F

Since sets of the form A

× ··· × A

generate R

, the desired result follows from

Theorem 1.3.1.

Theorem 1.3.4. If X

, . . . , X

are random variables then X

+ . . . + X

is a random

variable.

Proof. In view of Theorem 1.3.3 it suﬃces to show that f(x

, . . . , x

) = x

+ . . . + x

is measurable. To do this, we use Example 1.3.1 and note that {x : x

+ . . . + x

< a}

is an open set and hence is in R

14 CHAPTER 1. MEASURE THEORY

Theorem 1.3.5. If X

, X

, . . . are random variables then so are

inf

sup

lim sup

lim inf

Proof. Since the inﬁmum of a sequence is < a if and only if some term is < a (if all

terms are ≥ a then the inﬁmum is), we have

{inf

< a} = ∪

< a} ∈ F

A similar argument shows {sup

> a} = ∪

> a} ∈ F. For the last two, we

observe

lim inf

n→∞

= sup



inf

m≥n



lim sup

n→∞

= inf



sup

m≥n



To complete the proof in the ﬁrst case, note that Y

= inf

m≥n

is a random variable

for each n so sup

is as well.

From Theorem 1.3.5, we see that

Ω

≡ {ω : lim

n→∞

exists } = {ω : lim sup

n→∞

− lim inf

n→∞

= 0}

is a measurable set. (Here ≡ indicates that the ﬁrst equality is a deﬁnition.) If

P (Ω

) = 1, we say that X

converges almost surely, or a.s. for short. This type

of convergence called almost everywhere in measure theory. To have a limit deﬁned

on the whole space, it is convenient to let

∞

= lim sup

n→∞

but this random variable may take the value +∞ or −∞. To accommodate this and

some other headaches, we will generalize the deﬁnition of random variable.

A function whose domain is a set D ∈ F and whose range is R

∗

≡ [−∞, ∞] is said

to be a random variable if for all B ∈ R

∗

we have X

−1

(B) = {ω : X(ω) ∈ B} ∈ F.

Here R

∗

= the Borel subsets of R

∗

with R

∗

given the usual topology, i.e., the one

generated by intervals of the form [−∞, a), (a, b) and (b, ∞] where a, b ∈ R. The

reader should note that the extended real line (R

∗

, R

∗

) is a measurable space, so

all the results above generalize immediately.

Exercises

1.3.1. Show that if A generates S, then X

−1

(A) ≡ {{X ∈ A} : A ∈ A} generates

σ(X) = {{X ∈ B} : B ∈ S}.

1.3.2. Prove Theorem 1.3.4 when n = 2 by checking {X

+ X

< x} ∈ F.

1.3.3. Show that if f is continuous and X

→ X almost surely then f(X

) → f (X)

almost surely.

1.3.4. (i) Show that a continuous function from R

→ R is a measurable map from

, R

) to (R, R). (ii) Show that R

is the smallest σ-ﬁeld that makes all the

continuous functions measurable.

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