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 28
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... P[§:/;:?':;p 5 I] is the distribution function of the normalized number of ... continuity point x of F. Thus the De MoivreLaplace theorem says that (1) ... set {x} consisting of x alone has P-measure 0, Fn : F means that the implication (6) ...
... P[§:/;:?':;p 5 I] is the distribution function of the normalized number of ... continuity point x of F. Thus the De MoivreLaplace theorem says that (1) ... set {x} consisting of x alone has P-measure 0, Fn : F means that the implication (6) ...
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... continuity, and then choose a y in Dk so that - $(i/k)| < 6 for each i. Now ... p(x, y) S 25. And C is also complete: If xn is fundamental, which means that ... set is nowhere dense and that C is not o-compact. For 0 s t1 < < tk s 1 ...
... continuity, and then choose a y in Dk so that - $(i/k)| < 6 for each i. Now ... p(x, y) S 25. And C is also complete: If xn is fundamental, which means that ... set is nowhere dense and that C is not o-compact. For 0 s t1 < < tk s 1 ...
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... set A in 8 whose boundary 8A satisfies P(8A) = 0 is called a P-continuity set (note that 8A is closed and hence lies in S). Let P”, P be probability measures on (S, S). Theorem 2.1. These five conditions are equivalent: (i) Pn : P. (ii) ...
... set A in 8 whose boundary 8A satisfies P(8A) = 0 is called a P-continuity set (note that 8A is closed and hence lies in S). Let P”, P be probability measures on (S, S). Theorem 2.1. These five conditions are equivalent: (i) Pn : P. (ii) ...
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... P-continuity set except for countably many t. By condition (v) and the bounded convergence theorem, 1 I P,,f=/O Pn[f>t]dt—§/O P[f>t]dt=Pf. Other Criteria Weak convergence is often proved by showing that PnA —> PA holds for the sets A of ...
... P-continuity set except for countably many t. By condition (v) and the bounded convergence theorem, 1 I P,,f=/O Pn[f>t]dt—§/O P[f>t]dt=Pf. Other Criteria Weak convergence is often proved by showing that PnA —> PA holds for the sets A of ...
Pagina
... P-continuity sets in A satisfies the hypothesis of Theorem 2.3. For given A, let Ax,' be the class of A-sets ... set of P-measure 0. This means that each AI,' contains an element of AP, which therefore satisfies the hypothesis of Theorem ...
... P-continuity sets in A satisfies the hypothesis of Theorem 2.3. For given A, let Ax,' be the class of A-sets ... set of P-measure 0. This means that each AI,' contains an element of AP, which therefore satisfies the hypothesis of Theorem ...
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