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# 概率理论与例题习题解答（英文版－durrett）.pdf

Solutions Manual
The creation of this solution manual was one of the most important im-
provements in the second edition of Probability: Theory and Examples. The
solutions are not intended to be as polished as the proofs in the book, but are
supposed to give enough of the details so that little is left to the reader’s imag-
ination. It is inevitable that some of the many solutions will contain errors. If
you ﬁnd mistakes or better solutions send them via e-mail to ******@
or via post to Rick Durrett, Dept. of Math., 523 Malott Hall, Cornell U., Ithaca
NY 14853.
Rick Durrett
Contents
1 Laws of Large Numbers 1
1. Basic Deﬁnitions 1
2. Random Variables 3
3. Expected Value 4
4. Independence 7
5. Weak Laws of Large Numbers 12
6. Borel-Cantelli Lemmas 15
7. Strong Law of Large Numbers 19
8. Convergence of Random Series 20
9. Large Deviations 24
2 Central Limit Theorems 26
1. The De Moivre-Laplace Theorem 26
2. Weak Convergence 27
3. Characteristic Functions 31
4. Central Limit Theorems 35
6. Poisson Convergence 39
7. Stable Laws 43
8. Inﬁnitely Divisible Distributions 45
9. Limit theorems in Rd 46
3 Random walks 48
1. Stopping Times 48
4. Renewal theory 51
Contents iii
4 Martingales 54
1. Conditional Expectation 54
2. Martingales, Almost Sure Convergence 57
3. Examples 43
4. Doob’s Inequality, Lp Convergence 64
5. Uniform Integrability, Convergence in L1 66
6. Backwards Martingales 68
7. Optional Stopping Theorems 69
5 Markov Chains 74
1. Deﬁnitions and Examples 74
2. Extensions of the Markov Property 75
3. Recurrence and Transience 79
4. S