Convergence of Probability MeasuresJohn Wiley & Sons, 25 giu 2013 - 304 pagine A new look at weak-convergence methods in metric spaces-from a master of probability theory In this new edition, Patrick Billingsley updates his classic work Convergence of Probability Measures to reflect developments of the past thirty years. Widely known for his straightforward approach and reader-friendly style, Dr. Billingsley presents a clear, precise, up-to-date account of probability limit theory in metric spaces. He incorporates many examples and applications that illustrate the power and utility of this theory in a range of disciplines-from analysis and number theory to statistics, engineering, economics, and population biology. With an emphasis on the simplicity of the mathematics and smooth transitions between topics, the Second Edition boasts major revisions of the sections on dependent random variables as well as new sections on relative measure, on lacunary trigonometric series, and on the Poisson-Dirichlet distribution as a description of the long cycles in permutations and the large divisors of integers. Assuming only standard measure-theoretic probability and metric-space topology, Convergence of Probability Measures provides statisticians and mathematicians with basic tools of probability theory as well as a springboard to the "industrial-strength" literature available today. |
Dall'interno del libro
Risultati 1-5 di 34
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... subsets of the line, are uniquely determined by the requirements Pn(_°°s$I : Fn(m)a P(ToorxI : Since F is continuous ... subset A of the line; 8A consists of those points that are limits of sequences of points in A and are also limits of ...
... subsets of the line, are uniquely determined by the requirements Pn(_°°s$I : Fn(m)a P(ToorxI : Since F is continuous ... subset A of the line; 8A consists of those points that are limits of sequences of points in A and are also limits of ...
Pagina
... subsets A of C—Borel sets relative to the metric (11)—by P,,(A) = P[w:X"(w) 6 A] (the definition is possible because the mapping w —> X”(w) turns out to be measurable in the right way). In Chapter 2 we prove Donsker's theorem, which ...
... subsets A of C—Borel sets relative to the metric (11)—by P,,(A) = P[w:X"(w) 6 A] (the definition is possible because the mapping w —> X”(w) turns out to be measurable in the right way). In Chapter 2 we prove Donsker's theorem, which ...
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... subsets of the unit square. Chapter 4 concerns weak convergence of the distributions of random functions derived from various sequences of dependent random variables. Chapter 5 has to do with other asymptotic properties of random ...
... subsets of the unit square. Chapter 4 concerns weak convergence of the distributions of random functions derived from various sequences of dependent random variables. Chapter 5 has to do with other asymptotic properties of random ...
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... subsets K of A. Theorem 1.3. If S is separable and complete, then each probability measure on (8,8) is tight. PROOF. Since 8 is separable, there is, for each k, a sequence Am, Ak2, of open 1lK-balls covering 8. Choose nk large enough ...
... subsets K of A. Theorem 1.3. If S is separable and complete, then each probability measure on (8,8) is tight. PROOF. Since 8 is separable, there is, for each k, a sequence Am, Ak2, of open 1lK-balls covering 8. Choose nk large enough ...
Pagina
... subset consists of those points having only finitely many nonzero coordinates, each of them rational. If {x”} is fundamental, then each {1?} is fundamental and hence converges to some x,-, and of course x” converges to the point with ...
... subset consists of those points having only finitely many nonzero coordinates, each of them rational. If {x”} is fundamental, then each {1?} is fundamental and hence converges to some x,-, and of course x” converges to the point with ...
Sommario
THE SPACE C | ix |
THE SPACE D | viii |
DEPENDENT VARIABLES | xlviii |
OTHER MODES OF CONVERGENCE | lxiii |
APPENDIX_M | 16 |
INDEX | 38 |
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Analysis apply argument assume Borel o-field Borel sets Brownian motion cadlag functions central limit theorem choose compact set condition contains convergence in distribution convergence-determining class converges weakly convex countable defined denote dense density distribution function Donsker’s theorem equivalent ergodic Example exist finite finite-dimensional distributions finite-dimensional sets hence holds hypothesis image and image image image inequality infimum integral interval Lebesgue measure Lemma Let image lim sup limiting distribution linear log log mapping theorem measurable Image metric space nonnegative open sets P-continuity set partial sums permutation points polygonal positive prime divisors probability measure probability space proof of Theorem prove random element random function random variables random walk relatively compact satisfies Second Edition Section sequence Skorohod topology stationary Statistical subsequence subset Suppose supremum Theorem 3.1 tight uniformly continuous uniformly distributed values variance weak convergence Wiener measure