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 66
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... sequences of points in A and are also limits of sequences of points outside A. Since the boundary of (w, x] consists of the single point x, (6) is the same thing as (7)Pn(A) —* = I). where we have written A for (— w, x]. The fact of the ...
... sequences of points in A and are also limits of sequences of points outside A. Since the boundary of (w, x] consists of the single point x, (6) is the same thing as (7)Pn(A) —* = I). where we have written A for (— w, x]. The fact of the ...
Pagina
... sequence of independent, identically distributed random variables defined on some probability space (S2, .7, P). If the in have mean 0 and variance o2, then, by the LindebergLévy central limit theorem, the distribution of the normalized ...
... sequence of independent, identically distributed random variables defined on some probability space (S2, .7, P). If the in have mean 0 and variance o2, then, by the LindebergLévy central limit theorem, the distribution of the normalized ...
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... sequences of dependent random variables. Chapter 5 has to do with other asymptotic properties of random functions; there we prove Strassens's theorem, a far-reaching generalization of the law of the iterated logarithm. Many of the ...
... sequences of dependent random variables. Chapter 5 has to do with other asymptotic properties of random functions; there we prove Strassens's theorem, a far-reaching generalization of the law of the iterated logarithm. Many of the ...
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... sequence Am, Ak2, of open 1lK-balls covering 8. Choose nk large enough that P(UKM A“) > 1 — c/2k_ By the completeness hypothesis, the totally bounded set nk>1Ui<nk A!" has compact closure K. But clearly PK > 1 ¢ 6. Some Examples Here ...
... sequence Am, Ak2, of open 1lK-balls covering 8. Choose nk large enough that P(UKM A“) > 1 — c/2k_ By the completeness hypothesis, the totally bounded set nk>1Ui<nk A!" has compact closure K. But clearly PK > 1 ¢ 6. Some Examples Here ...
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... sequences x = (x1, x2,...) of real numbers—the product of countably many copies of R1. lf b(a, [3) = 1 A la — Bl, then b is a metric on R1 equivalent to the usual one, and under it, A'1 is complete as well as separable [M4]. Metrize R ...
... sequences x = (x1, x2,...) of real numbers—the product of countably many copies of R1. lf b(a, [3) = 1 A la — Bl, then b is a metric on R1 equivalent to the usual one, and under it, A'1 is complete as well as separable [M4]. Metrize R ...
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