概率不等式

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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