Exercises and Solutions Manual for Integration and Probability

I Measurable Spaces and Integrable Functions.- 1 ?-algebras and partitions.- 2 r-families.- 3 Monotone classes and independence.- 4 Banach limits.- 5 A strange probability measure.- 6 Integration and distribution functions.- 7 Evaluating $$ \sum\nolimits_{n = 1}^\infty {\frac{{{{\left( { - 1} \right)}^n}}}{{{n^2}}}} $$.- 8 Monotone convergence.- 9 Vector integration.- 10 Convergence in measure and composition of functions.- 11 Principle of separation of variables.- 12 The Cauchy-Schwarz inequality.- 13 Test that X ? Y almost everywhere.- 14 Image of a measure.- 15 Primitives of square integrable functions.- II Borel Measures and Radon Measures.- 1 Positive measures on an open interval.- 2 Distribution functions.- 3 Convexity and growth.- 4 Convexity and measure.- 5 Integral representation of positive convex functions (0, ?).- 6 Integral representations of Askey functions.- 7 Gauss's inequality.- 8 Integral of a decreasing function.- 9 Second mean value theorem for integrals.- 10 Variance of a distribution on [0, 1].- 11 Variance of the distribution of a convex function on [0, 1].- 12 Rational functions which preserve Lebesgue measure.- 13 A measure on the half-plane.- 14 Weak convergence and moments.- 15 Improper integrals and Lebesgue measure.- 16 $$ \int_0^\infty {\frac{{\sin x}}{x}} dx,\int_0^\infty {\left( {\cos ax - \cos bx} \right)} \frac{{dx}}{x},\int_0^\infty {\left( {\cos ax - \cos bx} \right)} \frac{{dx}}{{{x^2}}} $$.- 17 Comparisons between different Lp spaces.- 18 Differentiation under the integral sign.- 19 Laplace transform of a measure on [0,+?).- 20 Comparison of vague, weak, and narrow convergence.- 21 Weak compactness of measures.- 22 Vague convergence and limit of µn (0).- 23 Vague convergence and restriction to a closed set.- 24 Change of variables in an integral.- 25 Image of a measure and the Jacobian.- III Fourier Analysis.- 1 Characterizations of radial measures.- 2 Radial measures and independence.- 3 Area of the sphere.- 4 Fourier transform of the Poisson kernel of R+n+1.- 5 Askey-Polya functions.- 6 Symmetric convex sets in the plane and measures on [0,?).- 7 T. Ferguson's theorem.- 8 A counterexample of Herz.- 9 Riesz kernels.- 10 Measures on the circle and holomorphic functions.- 11 Harmonic polynomials and the Fourier transform.- 12 Bernstein's inequality.- 13 Cauchy's functional equation.- 14 Poisson's formula.- 15 A list of Fourier-Plancheral transforms.- 16 Fourier-Plancheral transform of a rational function.- 17 Computing some Fourier-Plancheral transforms.- 18 Expressing the Fourier-Plancheral transform as a limit.- 19 An identity for the Fourier-Plancheral transform.- 20 The Hilbert transform on L2(R).- 21 Action of L1(R) on L2(R).- 22 Another expression for the Hilbert transform.- 23 A table of properties of the Hilbert transform.- 24 Computing some Hilbert transforms.- 25 The Hilbert transform and distributions.- 26 Sobolev spaces on R.- 27 H. Weyl's inequality.- IV Hilbert Space Methods and Limit Theorems in Probability Theory.- 1 Fancy dice.- 2 The geometric distribution.- 3 The binomial and Poisson distributions.- 4 Construction of given distributions.- 5 Von Neumann's method.- 6 The laws of large numbers.- 7 Etemadi's method.- 8 A lemma on the random walks Sn.- 9 ?(s) = limn?? (P[Sn ? s·n])1/n exists.- 10 Evaluating ?(s) in some concrete cases.- 11 Algebra of the gamma and beta distributions.- 12 The gamma distribution and the normal distribution.- 13 The Cauchy distribution and the normal distribution.- 14 A probabilistic proof of Stirling's formula.- 15 Maxwell's theorem.- 16 If X1 and X2 are independent, then $$\frac{{\left( {{x_1},{x_2}} \right)}}{{{{\left( {x_1^2 + x_2^2} \right)}^{1/2}}}}$$ is uniform.- 17 Isotropy of pairs and triplets of independent variables.- 18 The only invertible distributions are concentrated at a point.- 19 Isotropic multiples of normal distributions.- 20 Poincaré's lemma.- 21 Schoenberg's theorem.- 22 A property of radial distributions.- 23 Brownian motion hits a hyperplane in a Cauchy distribution.- 24 Pittinger's inequality.- 25 Cylindrical probabilities.- 26 Minlos's lemma.- 27 Condition that a cylindrical probability be a probability measure.- 28 Lindeberg's theorem.- 29 H. Chernoff's inequality.- 30 Gebelein's inequality.- 31 Fourier transform of the Hermite polynomials.- 32 Another definition of conditional expectation.- 33 Monotone continuity of conditional expectations.- 34 Concrete computation of conditional expectations.- 35 Conditional expectations and independence.- 36 E(X|Y) = Y and E(Y|X) = X.- 37 Warnings about conditional expectations.- 38 Conditional expectations in the absolutely continuous case and the Gaussian case.- 39 Examples of martingales.- 40 A reversed martingale.- 41 A probabilistic approximation of an arithmetic conjecture.- 42 A criterion for uniform integrability.- 43 The Galton-Watson process and martingales.- V Gaussian Sobolev Spaces and Stochastic Calculus of Variations.- 1 d and ? cannot both be continuous.- 2 Growth of the Hermite polynomials.- 3 Viskov's lemma.- 4 Cantelli's conjecture.- 5 Lancaster probabilities in R2.- 6 Sarmanov's theorem.