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 83
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... Theorem 1.1 implies that P is completely determined by the values of PF for closed sets F. The next theorem shows that P is also determined by the values of Pf for bounded, continuous f. The proof depends on approximating the indicator ...
... Theorem 1.1 implies that P is completely determined by the values of PF for closed sets F. The next theorem shows that P is also determined by the values of Pf for bounded, continuous f. The proof depends on approximating the indicator ...
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Patrick Billingsley. The Portmanteau Theorem The following theorem provides useful conditions equivalent to weak ... PROOF of Theorem 2.1. Of course, the implication (i) —> (ii) is trivial. Proof that (ii) —> (iii). The fof (1.1) is ...
Patrick Billingsley. The Portmanteau Theorem The following theorem provides useful conditions equivalent to weak ... PROOF of Theorem 2.1. Of course, the implication (i) —> (ii) is trivial. Proof that (ii) —> (iii). The fof (1.1) is ...
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... theorem, 1 I P,,f=/O Pn[f>t]dt—§/O P[f>t]dt=Pf. Other Criteria Weak ... PROOF. lf A1,... A, lie in AP, then so do their intersections; hence, by the ... theorem holds. The next result transforms condition (ii) above in a useful way ...
... theorem, 1 I P,,f=/O Pn[f>t]dt—§/O P[f>t]dt=Pf. Other Criteria Weak ... PROOF. lf A1,... A, lie in AP, then so do their intersections; hence, by the ... theorem holds. The next result transforms condition (ii) above in a useful way ...
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... Theorem 2.3. For given A, let Ax,' be the class of A-sets satisfying 3: 6 A ... PROOF. Fix an arbitrary P and let AP be the class of P-continuity sets in A ... Theorem 2.3. Since PnA —> PA for each A in AP, it follows that Pn => P ...
... Theorem 2.3. For given A, let Ax,' be the class of A-sets satisfying 3: 6 A ... PROOF. Fix an arbitrary P and let AP be the class of P-continuity sets in A ... Theorem 2.3. Since PnA —> PA for each A in AP, it follows that Pn => P ...
Pagina
... Theorem 2.2 has a corollary used in Section 4. Recall that A is a semiring ... PROOF. lfA1, A, lie in A, then, since A is a semiring, Urn A. can be ... proof is completed as before. A further simple condition for weak convergence: Theorem ...
... Theorem 2.2 has a corollary used in Section 4. Recall that A is a semiring ... PROOF. lfA1, A, lie in A, then, since A is a semiring, Urn A. can be ... proof is completed as before. A further simple condition for weak convergence: 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