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 143
... convolution operation ☆ is com- mutative and associative . Theorem 4. If G is continuous ( = free of atoms ) , so is U = F ☆ G . If G has the ordinary density q , then U has the ordinary density u given by ( 4.1 ) . ( Because of ...
... convolution operation ☆ is com- mutative and associative . Theorem 4. If G is continuous ( = free of atoms ) , so is U = F ☆ G . If G has the ordinary density q , then U has the ordinary density u given by ( 4.1 ) . ( Because of ...
Page 288
William Feller. Corollary . Distinct convolution semi - groups cannot have the same gen- erator . Lemma 2. ( Convergence ) . For each n let An generate the convolution semi - group { Q , ( t ) } . An If A , → A , then A generates a ...
William Feller. Corollary . Distinct convolution semi - groups cannot have the same gen- erator . Lemma 2. ( Convergence ) . For each n let An generate the convolution semi - group { Q , ( t ) } . An If A , → A , then A generates a ...
Page 411
... Convolutions . Let F and G be probability distributions and U their convolution , that is , ( 2.1 ) U ( x ) = [ " G ( x − y ) F { dy } ; The corresponding Laplace transforms obey the multiplication rule ( 2.2 ) ω = φγ . This is ...
... Convolutions . Let F and G be probability distributions and U their convolution , that is , ( 2.1 ) U ( x ) = [ " G ( x − y ) F { dy } ; The corresponding Laplace transforms obey the multiplication rule ( 2.2 ) ω = φγ . This is ...
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