概率论教程

preface

1 basic measure theory

1.1 classes of sets

1.2 set functior

1.3 the measure exterion theorem

1.4 measurable maps

1.5 random variables

2 independence

2.1 independence of events

2.2 independent random variables

2.3 kolmogorov's 0-1 law

2.4 example：percolation

3 generating functior

3.1 definition and examples

3.2 poisson approximation

3.3 branching processes

4 the integral

4.1 cortruction and simple properties

4.2 monotone convergence and fatou's lemma

.4.3 lebesgue integral verus riemann integral

5 moments and laws of large number

5.1 moments

5.2 weak law of large number

5.3 strong law of large number

5.4 speed of convergence in the strong lln

5.5 the poisson process

6 convergence theorems

6.1 almost sure and measure convergence

6.2 uniform integrability

6.3 exchanging integral and differentiation

7 lp-spaces and the radon-nikodym theorem

7.1 definitior

7.2 inequalities and the fischer-riesz theorem

7.3 hilbert spaces

7.4 lebesgue's decomposition theorem

7.5 supplement：signed measures

7.6 supplement：dual spaces

8 conditional expectatior

8.1 elementary conditional probabilities

8.2 conditional expectatior

8.3 regular conditional distribution

9 martingales

9.1 processes, filtratior, stopping times

9.2 martingales

9.3 discrete stochastic integral

9.4 discrete martingale representation theorem and the crr model

10 optional sampling theorems

10.1 doob decomposition and square variation

10.2 optional sampling and optional stopping

10.3 uniform integrability and optional sampling

11 martingale convergence theorems and their applicatior

11.1 doob's inequality

11.2 martingale convergence theorems

11.3 example：branching process

12 backwards martingales and exchangeability

12.1 exchangeable families of random variables

12.2 backwards martingales

12.3 de finetti's theorem

13 convergence of measures

13.1 a topology primer

13.2 weak and vague convergence

13.3 prohorov's theorem

13.4 application：a fresh look at de finetti's theorem

14 probability measures on product spaces

14.1 product spaces

14.2 finite products and trarition kernels

14.3 kolmogorov's exterion theorem

14.4 markov semigroups

15 characteristic functior and the central limit theorem

15.1 separating classes of functior

15.2 characteristic functior：examples

15.3 l6vy's continuity theorem

15.4 characteristic functior and moments

15.5 the central limit theorem

15.6 multidimerional central limit theorem

16 infinitely divisible distributior

16.1 l6vy-khinchin formula

16.2 stable distributior

17 markov chair

17.1 definitior and cortruction

17.2 discrete markov chair：examples

17.3 discrete markov processes in continuous time

17.4 discrete markov chair：recurrence and trarience

17.5 application：recurrence and trarience of random walks

17.6 invariant distributior

18 convergence of markov chair

18.1 periodicity of markov chair

18.2 coupling and convergence theorem

18.3 markov chain monte carlo method

18.4 speed of convergence

19 markov chair and electrical networks

19.1 harmonic functior

19.2 reverible markov chair

19.3 finite electrical networks

19.4 recurrence and trarience

19.5 network reduction

19.6 random walk in a random environment

20 ergodic theory

20.1 definitior

20.2 ergodic theorems

20.3 examples

20.4 application：recurrence of random walks

20.5 mixing

21 brownian motion

21.1 continuous verior

21.2 cortruction and path properties

21.3 strong markov property

21.4 supplement：feller processes

21.5 cortruction via l2-approximation

21.6 the space c（[0, ∞））

21.7 convergence of probability measures on c（[0, ∞））

21.8 dorker's theorem

21.9 pathwise convergence of branching processes

21.10 square variation and local martingales

22 law of the iterated logarithm

22.l iterated logarithm for the brownian motion

22.2 skorohod's embedding theorem

22.3 hartman-wintner theorem

23 large deviatior

23.1 cramer's theorem

23.2 large deviatior principle

23.3 sanov's theorem

23.4 varadhan's lemma and free energy

24 the poisson point process

24.1 random measures

24.2 properties of the poisson point process

24.3 the poisson-dirichlet distribution

25 the it6 integral

25.1 it6 integral with respect to brownian motion

25.2 it6 integral with respect to diffusior

25.3 the it6 formula

25.4 dirichlet problem and brownian motion

25.5 recurrence and trarience of brownian motion

26 stochastic differential equatior

26.1 strong solutior

26.2 weak solutior and the martingale problem

26.3 weak uniqueness via duality

references

notation index

name index

subject index

《概率论教程》是一部讲述现代概率论及其测度论应用基础的教程，其目标读者是该领域的研究生和相关的科研人员。内容广泛，有许多初级教程不能涉及到得的。理论叙述严谨，独立性强。有关测度的部分和概率的章节相互交织，将概率的抽象性完全呈现出来。此外，还有大量的图片、计算模拟、重要数学家的个人传记和大量的例子。这使得表现形式更加活跃。