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 29
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... tight if for each 6 there exists a compact set K such that PK > 1 _ 6. By Theorem 1.1, P is tight if and only if PA is, for each A in S, the supremum of PK over the compact subsets K of A. Theorem 1.3. If S is separable and complete ...
... tight if for each 6 there exists a compact set K such that PK > 1 _ 6. By Theorem 1.1, P is tight if and only if PA is, for each A in S, the supremum of PK over the compact subsets K of A. Theorem 1.3. If S is separable and complete ...
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... tight. But tightness is in this case obvious because the space is o-compact—is a countable union of compact sets (B(O, n)- T R'', for example). Example 1.2. Let R°° be the space of sequences x = (x1, x2,...) of real numbers—the product ...
... tight. But tightness is in this case obvious because the space is o-compact—is a countable union of compact sets (B(O, n)- T R'', for example). Example 1.2. Let R°° be the space of sequences x = (x1, x2,...) of real numbers—the product ...
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... tight. Like R”, C is not o-compact. To see this, consider the function (1.4) again. No subsequence of 62-,' can converge, because if p(¢gm,z) -i 0' then 2 must be the 0function, while p(czm,0) = 6. Therefore, the closed ball B(0, 6)_ is ...
... tight. Like R”, C is not o-compact. To see this, consider the function (1.4) again. No subsequence of 62-,' can converge, because if p(¢gm,z) -i 0' then 2 must be the 0function, while p(czm,0) = 6. Therefore, the closed ball B(0, 6)_ is ...
Pagina
Patrick Billingsley. Suppose there exists on S a probability measure P that is not tight. If So consists of all the x for which P{x} > 0, then SO is countablei and hence PSO < 1 (since othen/vise P would be tight). But then, if MA : P(A ...
Patrick Billingsley. Suppose there exists on S a probability measure P that is not tight. If So consists of all the x for which P{x} > 0, then SO is countablei and hence PSO < 1 (since othen/vise P would be tight). But then, if MA : P(A ...
Pagina
... tight. 1.14. If 8 consists of the rationals with the relative topology of the line, then each P on S is tight, even though 8 is not topologically complete (use the Baire category theorem). 1.15. If r s ¢/(1+ 5)? , then (see (1.3)) B(g ...
... tight. 1.14. If 8 consists of the rationals with the relative topology of the line, then each P on S is tight, even though 8 is not topologically complete (use the Baire category theorem). 1.15. If r s ¢/(1+ 5)? , then (see (1.3)) B(g ...
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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