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Author Letac, Gรฉrard. author Exercises and Solutions Manual for Integration and Probability [electronic resource] : by Paul Malliavin / by Gรฉrard Letac New York, NY : Springer New York, 1995 http://dx.doi.org/10.1007/978-1-4612-4212-3 142 p. 1 illus. online resource

SUMMARY

This book presents the problems and worked-out solutions for all the exercises in the text by Malliavin. It will be of use not only to mathematics teachers, but also to students using the text for self-study

CONTENT

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 $$\sum\nolimits_{n = 1}̂\infty {\frac{{{{\left( { - 1} \right)}̂n}}}{{{n̂2}}}}$$ -- 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 Gaussโ{128}{153}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 $$\int_0̂\infty {\frac{{\sin x}}{x}} dx,\int_0̂\infty {\left( {\cos ax - \cos bx} \right)} \frac{{dx}}{x},\int_0̂\infty {\left( {\cos ax - \cos bx} \right)} \frac{{dx}}{{{x̂2}}}$$ -- 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 ยตn (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. Fergusonโ{128}{153}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 Bernsteinโ{128}{153}s inequality -- 13 Cauchyโ{128}{153}s functional equation -- 14 Poissonโ{128}{153}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. Weylโ{128}{153}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 Neumannโ{128}{153}s method -- 6 The laws of large numbers -- 7 Etemadiโ{128}{153}s method -- 8 A lemma on the random walks Sn -- 9 ?(s) = limn?? (P[Sn ? sยทn])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 Stirlingโ{128}{153}s formula -- 15 Maxwellโ{128}{153}s theorem -- 16 If X1 and X2 are independent, then $$\frac{{\left( {{x_1},{x_2}} \right)}}{{{{\left( {x_1̂2 + x_2̂2} \right)}̂{1/2}}}}$$ 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 Poincarรฉโ{128}{153}s lemma -- 21 Schoenbergโ{128}{153}s theorem -- 22 A property of radial distributions -- 23 Brownian motion hits a hyperplane in a Cauchy distribution -- 24 Pittingerโ{128}{153}s inequality -- 25 Cylindrical probabilities -- 26 Minlosโ{128}{153}s lemma -- 27 Condition that a cylindrical probability be a probability measure -- 28 Lindebergโ{128}{153}s theorem -- 29 H. Chernoffโ{128}{153}s inequality -- 30 Gebeleinโ{128}{153}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 and E(Y ) = 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 Viskovโ{128}{153}s lemma -- 4 Cantelliโ{128}{153}s conjecture -- 5 Lancaster probabilities in R2 -- 6 Sarmanovโ{128}{153}s theorem

Mathematics Functional analysis Measure theory Probabilities Mathematics -- Study and teaching Mathematics Probability Theory and Stochastic Processes Measure and Integration Functional Analysis Mathematics Education

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