About This Course

This book rebuilds probability on measure theory and then uses that foundation to prove the limit theorems that a first course can only state. It is written for a reader working alone at the level of a first-year graduate course: someone who has computed with distributions and expectations in Probability Theory, who has met sigma-algebras, the Lebesgue integral, the convergence theorems, product measures, \(L^p\) spaces, and the Radon–Nikodym theorem in Measure Theory, and who writes and reads proofs with the fluency that Real Analysis develops.

The division of labour between this book and its prerequisites is strict. Measure Theory owns the general theory of measures and integrals; whenever one of its theorems is used here, it is named, the hypotheses the argument needs are stated at the point of use, and the result is marked once with its library identifier. It is not proved again. Probability Theory owns the classical statements of the laws of large numbers and of the central limit theorem and their use in calculation; this book owns their proofs at full generality. Processes in continuous time, Brownian motion, and stochastic integration belong to later courses and appear here only as destinations.

The book has done its work if, with the cover closed, you can carry out each of the following. Translate a probabilistic statement into a statement about measurable maps, sigma-algebras, and laws, and say which of the three it is really about. Decide whether events happen infinitely often and whether a tail event is trivial. Tell the four modes of convergence apart, prove each implication among them, and produce the standard counterexample for each implication that fails. Decide when expectations pass to the limit by testing uniform integrability. Compute and manipulate conditional expectations from their defining property alone. Verify that a process is a martingale, stop it legitimately, and prove that it converges. Prove the weak and strong laws of large numbers and the central limit theorem, and choose among all of these tools when a new sequence is put in front of you.

Notation is conventional. A probability space is written \((\Omega,\mathcal F,\mathbb P)\) and expectation is \(\mathbb E\) . The indicator of an event \(A\) is \(\mathbf 1_A\) , and \(\mathbb E[X;A]\) abbreviates \(\mathbb E[X\mathbf 1_A]\) . The Borel sigma-algebra of \(\mathbb R\) is \(\mathcal B(\mathbb R)\) and Lebesgue measure is \(\lambda\) . For real numbers, \(a\wedge b\) is the minimum, \(a\vee b\) the maximum, \(x^{+}=x\vee 0\) , and \(x^{-}=(-x)\vee 0\) . The abbreviation a.s. means almost surely, that is, outside an event of probability zero, and i.i.d. means independent and identically distributed. The norm of \(L^p\) is \(\lVert X\rVert_p=(\mathbb E\lvert X\rvert^p)^{1/p}\) .

Most of the exercises ask for proofs, and the companion solutions answer in kind: a full solution is a complete written argument, a hint gives the decisive step, and an answer gives only the result to compare with. Many chapters also contain exercises that present a plausible wrong argument and ask where it breaks. Probability is unusually rich in such arguments, because intuition formed on finitely many events fails for infinitely many, and learning to locate the failure is part of the subject. The cumulative review at the end does not say which chapter each problem belongs to; choosing the tool is the problem.