An Introduction to Probability Theory and Its Applications, Volume 2Wiley, 1950 - Probabilities Vol. 2 has series: Wiley series in probability and mathematical statistics. Bibliographical footnotes. "Some books on cagnate subjects": v. 2, p. 615-616. |
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Page 246
... implies the other and ( 2.1 ) . [ It is essential that none of the distributions be concentrated at one point . For example , let V be concentrated at the origin and F1 = U for all n . The first relation in ( 2.2 ) holds with a , = 1 ...
... implies the other and ( 2.1 ) . [ It is essential that none of the distributions be concentrated at one point . For example , let V be concentrated at the origin and F1 = U for all n . The first relation in ( 2.2 ) holds with a , = 1 ...
Page 420
... implies the other two . Proof . We know already that ( 5.5 ) implies ( 5.6 ) and ( 5.6 ) implies ( 5.7 ) . To show that ( 5.7 ) implies ( 5.5 ) we repeat the preceding argument . To U ( tx ) there corresponds the Laplace transform ( 7λ ) ...
... implies the other two . Proof . We know already that ( 5.5 ) implies ( 5.6 ) and ( 5.6 ) implies ( 5.7 ) . To show that ( 5.7 ) implies ( 5.5 ) we repeat the preceding argument . To U ( tx ) there corresponds the Laplace transform ( 7λ ) ...
Page 425
... implies that the possible limits G in ( 6.3 ) differ only by scale factors from some G. It follows , in particular , that there are no other stable distributions concentrated on 0 , ∞ . Proof . If y and y are the Laplace transforms of ...
... implies that the possible limits G in ( 6.3 ) differ only by scale factors from some G. It follows , in particular , that there are no other stable distributions concentrated on 0 , ∞ . Proof . If y and y are the Laplace transforms of ...
Contents
CHAPTER | 1 |
SPECIAL DENSITIES RANDOMIZATION | 44 |
PROBABILITY MEASURES AND SPACES | 101 |
Copyright | |
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An Introduction to Probability Theory and Its Applications, Volume 2 William Feller Limited preview - 1991 |
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a₁ applies arbitrary argument assume asymptotic atoms backward equation Baire functions Borel sets bounded central limit theorem characteristic function common distribution compound Poisson condition consider constant continuous function convergence convolution defined definition denote density derived distribution F distribution function equals example exists exponential distribution F{dx finite interval fixed follows formula given hence implies independent random variables inequality infinitely divisible integral integrand Laplace transform law of large left side lemma Let F limit distribution Markov martingale measure mutually independent normal distribution notation o-algebra obvious operator parameter Poisson process positive probabilistic probability distribution problem proof prove random walk renewal epochs renewal equation renewal process S₁ sample space satisfies semi-group sequence shows solution stable distributions stochastic stochastic kernel symmetric T₁ tends theory transition probabilities uniformly unique variance vector X₁ Y₁ zero expectation