所属分类：数学

出版时间：1970-1 出版时间：科学出版社 作者：Zhengyan Lin, Zhidong Bai 页数：181

**前言**

In almost every branch of quantitative sciences， inequalities play an im-portant role in its development and are regarded to be even more impor-tant than equalities. This is indeed the case in probability and statis-tics. For example， the Chebyshev， Schwarz and Jensen inequalities arefrequently used in probability theory， the Cramer-Rao inequality playsa fundamental role in mathematical statistics. Choosing or establishingan appropriate inequality is usually a key breakthrough in the solutionof a problem， e.g. the Berry-Esseen inequality opens a way to evaluatethe convergence rate of the normal approximation. Research beginners usually face two difficulties when they start resear-ching——they choose an appropriate inequality and/or cite an exact ref-erence. In literature， almost no authors give references for frequentlyused inequalities， such as the Jensen inequality， Schwarz inequality， Fa-tou Lemma， etc. Another annoyance for beginners is that an inequalitymay have many different names and reference sources. For example，the Schwarz inequality is also called the Cauchy， Cauchy-Schwarz orMinkovski-Bnyakovski inequality. Bennet， Hoeffding and Bernstein in-equalities have a very close relationship and format， and in literaturesome authors cross-cite in their use of the inequalities. This may be dueto one author using an inequality and subsequent authors just simplycopying the inequalitys format and its reference without checking theoriginal reference. All this may distress beginners very much. The aim of this book is to help beginners with these problems. Weprovide a place to find the most frequently used inequalities， their proofs（if not too lengthy） and some references. Of course， for some of the morepopularly known inequalities， such as Jensen and Schwarz， there is nonecessity to give a reference and we will not do so.

**内容概要**

Inequality has become an essential tool in many areas of mathematical research, for example in probability and statistics where it is frequently used in the proofs. Probability Inequalities covers inequalities related with events, distribution functions, characteristic functions, moments and random variables (elements) and their sum. The book shall serve as a useful tool and reference for scientists in the areas of probability and statistics, and applied mathematics.

**作者简介**

Prof. Zhengyan Lin is a fellow of the Institute of Mathematical Statistics and currently a professor at Zhejiang University， Hangzhou， China. He is the prize winner of National Natural Science Award of China in 1997. Prof. Zhidong Bai is a fellow of TWAS and the Institute of Mathematical Statistics; he is a professor at the National University of Singapore and Northeast Normal University， Changchun， China.

**书籍目录**

Chapter

1

Elementary

Inequalities

of

Probabilities

of

Events

1.1

Inclusion-exclusion

Formula

1.2

Corollaries

of

the

Inclusion-exclusion

Formula

1.3

Further

Consequences

of

the

Inclusion-exclusion

Formula

1.4

Inequalities

Related

to

Symmetric

Difference

1.5

Inequalities

Related

to

Independent

Events

1.6

Lower

Bound

for

Union

(Chung-ErdSs)

ReferencesChapter

2

Inequalities

Related

to

Commonly

Used

Distributions

2.1

Inequalities

Related

to

the

Normal

d.f.

2.2

Slepian

Type

Inequalities

2.3

Anderson

Type

Inequalities

2.4

Khatri-Sidak

Type

Inequalities

2.5

Corner

Probability

of

Normal

Vector

2.6

Normal

Approximations

of

Binomial

and

Poisson

Distributions

ReferencesChapter

3

Inequalities

Related

to

Characteristic

Functions

3.1

Inequalities

Related

Only

with

c.f

3.2

Inequalities

Related

to

c.f.

and

d.f.

3.3

Normality

Approximations

of

c.f.

of

Independent

Sums

ReferencesChapter

4

Estimates

of

the

Difference

of

Two

Distribution

Functions

4.1

Fourier

Transformation

4.2

Stein-Chen

Method

4.3

Stieltjes

Transformation

ReferencesChapter

5

Probability

Inequalities

of

Random

Variables

5.1

Inequalities

Related

to

Two

r.v.'s

5.2

Perturbation

Inequality

5.3

Symmetrization

Inequalities

5.4

Levy

Inequality

5.5

Bickel

Inequality

5.6

Upper

Bounds

of

Tail

Probabilities

of

Partial

Sums

5.7

Lower

Bounds

of

Tail

Probabilities

of

Partial

Sums

5.8

Tail

Probabilities

for

Maximum

Partial

Sums

5.9

Tail

Probabilities

for

Maximum

Partial

Sums

(Continuation)

5.10

Reflection

Inequality

of

Tail

Probability

(HoffmannJorgensen)

5.11

Probability

of

Maximal

Increment

(Shao)

5.12

Mogulskii

Minimal

Inequality

5.13

Wilks

Inequality

ReferencesChapter

6

Bounds

of

Probabilities

in

Terms

of

Moments

6.1

Chebyshev-Markov

Type

Inequalities

6.2

Lower

Bounds

6.3

Series

of

Tail

Probabilities

6.4

Kolmogorov

Type

Inequalities

6.5

Generalization

of

Kolmogorov

Inequality

for

a

Submartingale

6.6

Renyi-Hajek

Type

Inequalities

6.7

Chernoff

Inequality

6.8

Fuk

and

Nagaev

Inequality

6.9

Burkholder

Inequality

6.10

Complete

Convergence

of

Partial

Sums

ReferencesChapter

7

Exponential

Type

Estimates

of

Probabilities

7.1

Equivalence

of

Exponential

Estimates

7.2

Petrov

Exponential

Inequalities

7.3

Hoeffding

Inequality

7.4

Bennett

Inequality

7.5

Bernstein

Inequality

7.6

Exponential

Bounds

for

Sums

of

Bounded

Variables

7.7

Kolmogorov

Inequalities

7.8

Prokhorov

Inequality

7.9

Exponential

Inequalities

by

Censoring

7.10

Tail

Probability

of

Weighted

Sums

ReferencesChapter

8

Moment

Inequalities

Related

to

One

or

Two

Variables

8.1

Moments

of

Truncation

8.2

Exponential

Moment

of

Bounded

Variables

8.3

HSlder

Type

Inequalities

8.4

Jensen

Type

Inequalities

8.5

Dispersion

Inequality

of

Censored

Variables

8.6

Monotonicity

of

Moments

of

Sums

8.7

Symmetrization

Moment

Inequatilies

8.8

Kimball

Inequality

8.9

Exponential

Moment

of

Normal

Variable

8.10

Inequatilies

of

Nonnegative

Variable

8.11

Freedman

Inequality

8.12

Exponential

Moment

of

Upper

Truncated

Variables

ReferencesChapter

9

Moment

Estimates

of

(Maximum

of)

Sums

of

Random

Variables

9.1

Elementary

Inequalities

9.2

Minkowski

Type

Inequalities

9.3

The

Case

1≤r≤2

9.4

The

Case

r≥2

9.5

Jack-knifed

Variance

9.6

Khintchine

Inequality

9.7

Marcinkiewicz-Zygmund-Burkholder

Type

Inequalities

9.8

Skorokhod

Inequalities

9.9

Moments

of

Weighted

Sums

9.10

Doob

Crossing

Inequalities

9.11

Moments

of

Maximal

Partial

Sums

9.12

Doob

Inequalities

9.13

Equivalence

Conditions

for

Moments

9.14

Serfiing

Inequalities

9.15

Average

Fill

Rate

ReferencesChapter

10

Inequalities

Related

to

Mixing

Sequences.

10.1

Covariance

Estimates

for

Mixing

Sequences

10.2

Tail

Probability

on

α-mixing

Sequence

10.3

Estimates

of

4-th

Moment

on

p-mixing

Sequence

10.4

Estimates

of

Variances

of

Increments

of

p-mixing

Sequence

10.5

Bounds

of

2+δ-th

Moments

of

Increments

of

p-mixing

Sequence

10.6

Tail

Probability

on

g-mixing

Sequence

10.7

Bounds

of

2+δ-th

Moment

of

Increments

of

mixing

Sequence

10.8

Exponential

Estimates

of

Probability

on

mixing

Sequence

ReferencesChapter

11

Inequalities

Related

to

Associative

Variables

11.1

Covariance

of

PQD

Varalbles

11.2

Probability

of

Quadrant

on

PA

(NA)

Sequence

11.3

Estimates

of

c.f.'s

on

LPQD

(LNQD)

Sequence

11.4

Maximal

Partial

Sums

of

PA

Sequence

11.5

Variance

of

Increment

of

LPQD

Sequence

11.6

Expectation

of

Convex

Function

of

Sum

of

NA

Sequence

11.7

Marcinkiewicz-Zygmund-Burkholder

Inequality

for

NA

Sequence

ReferencesChapter

12

Inequalities

about

Stochastic

Processes

and

Banach

Space

Valued

Random

Variables

12.1

Probability

Estimates

of

Supremums

of

a

Wiener

Process

12.2

Probability

Estimate

of

Supremum

of

a

Poisson

Process

12.3

Fernique

Inequality

12.4

Borell

Inequality

12.5

Tail

Probability

of

Gaussian

Process

12.6

Tail

Probability

of

Randomly

Signed

Independent

Processes

12.7

Tail

Probability

of

Adaptive

Process

12.8

Tail

Probability

on

Submartingale

12.9

Tail

Probability

of

Independent

Sum

in

B-Space

12.10

Isoperimetric

Inequalities

12,11

Ehrhard

Inequality

12.12

Tail

Probability

of

Normal

Variable

in

B-Space

12.13

Gaussian

Measure

on

Symmetric

Convex

Sets

12.14

Equivalence

of

Moments

of

B-Gaussian

Variables

12.15

Contraction

Principle

12.16

Symmetrization

Inequalities

in

B-Space

12.17

DecoupIing

Inequality

References

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