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 85
Pagina
... holds at every continuity point x of F. Thus the De MoivreLaplace theorem says that (1) converges weakly to (2); since (2) is everywhere continuous, the proviso about continuity points is vacuous in this case. If F,, and F are defined ...
... holds at every continuity point x of F. Thus the De MoivreLaplace theorem says that (1) converges weakly to (2); since (2) is everywhere continuous, the proviso about continuity points is vacuous in this case. If F,, and F are defined ...
Pagina
... holds for all bounded, continuous real-valued functions on S. (In order to conform with general mathematical usage, we take (10) as the defintion of weak convergence, so that (7) becomes a necessary and sufficient condition instead of a ...
... holds for all bounded, continuous real-valued functions on S. (In order to conform with general mathematical usage, we take (10) as the defintion of weak convergence, so that (7) becomes a necessary and sufficient condition instead of a ...
Pagina
... holds in the nonseparable case. The ball o-field will play a role only in Sections 6 and 15. Problems A simple ... holds for a continuous f but not for a uniformly continuous one. 1.4. If S is a Banach space, then either (i) no closed ...
... holds in the nonseparable case. The ball o-field will play a role only in Sections 6 and 15. Problems A simple ... holds for a continuous f but not for a uniformly continuous one. 1.4. If S is a Banach space, then either (i) no closed ...
Pagina
... holds because the set there consists of those k satisfying [010"] < k S ]_b10"J_ That Pn => P holds in this case is an expression of the fact that one can produce approximately uniformly distributed observations by generating a stream ...
... holds because the set there consists of those k satisfying [010"] < k S ]_b10"J_ That Pn => P holds in this case is an expression of the fact that one can produce approximately uniformly distributed observations by generating a stream ...
Pagina
... holds for intervals J, then by part (v) of the theorem, it also holds for a much wider class of sets—those having boundary of Lebesgue measure 0. PROOF of Theorem 2.1. Of course, the implication (i) —> (ii) is trivial. Proof that (ii) ...
... holds for intervals J, then by part (v) of the theorem, it also holds for a much wider class of sets—those having boundary of Lebesgue measure 0. PROOF of Theorem 2.1. Of course, the implication (i) —> (ii) is trivial. Proof that (ii) ...
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