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 248
... lemma can be proved in many ways ( see problem 4 ) and is given here chiefly to illustrate the use of the preceding lemmas . 1 ... Lemma 3. Let X1 , X2 , . . . be independent random variables with a common distribution F and S1 = X1 ++ ...
... lemma can be proved in many ways ( see problem 4 ) and is given here chiefly to illustrate the use of the preceding lemmas . 1 ... Lemma 3. Let X1 , X2 , . . . be independent random variables with a common distribution F and S1 = X1 ++ ...
Page 300
... lemma 1. In view of the symmetrization inequality V , ( 5.8 ) this implies that Xx.n lemma 2 applies to the array { Xx.nμn } . Thus n - 1S , — μn n - p μ - μn and hence P 0 . Since also n1s 0 the last relation implies → 0 , which ...
... lemma 1. In view of the symmetrization inequality V , ( 5.8 ) this implies that Xx.n lemma 2 applies to the array { Xx.nμn } . Thus n - 1S , — μn n - p μ - μn and hence P 0 . Since also n1s 0 the last relation implies → 0 , which ...
Page 529
... Lemma 1. If cx2 F , { dx } → M { dx } properly , then ( 1.18 ) Cn ∞- ∞ + . ° z ( x ) F „ { dx } → S * ∞- ∞ + . x2x ( x ) M { dx } for every bounded continuous function z such that x ̄2z ( x ) is continuous at the origin . Proof ...
... Lemma 1. If cx2 F , { dx } → M { dx } properly , then ( 1.18 ) Cn ∞- ∞ + . ° z ( x ) F „ { dx } → S * ∞- ∞ + . x2x ( x ) M { dx } for every bounded continuous function z such that x ̄2z ( x ) is continuous at the origin . Proof ...
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