Probability: Theory and Examples
This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The fourth edition begins with a short chapter on measure theory to orient readers new to the subject.
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apply Theorem Borel Borel-Cantelli lemma Brownian motion central limit theorem ch.f characteristic function completes the proof compute conclude conditional expectation continuous convergence theorem implies countable deﬁne defined deﬁnition density desired result follows disjoint distribution function distribution with mean dominated convergence theorem Example Exercise finite ﬁrst follows from Theorem formula Fubini's theorem inequality irreducible large numbers last result law of large Lebesgue measure let Sn lim sup liminf log log Markov chain Markov property martingale normal distribution observe P(Sn P(Xn permutation Poisson distribution Poisson process probability measure proof of Theorem prove the result random walk recurrent Remark right-hand side Section Show simple random walk space stationary distribution stationary measure stationary sequence stopping submartingale supermartingale Suppose transition probability trivial uniformly integrable variance