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 84
... positive definite . The converse is trivial when r = 1 and we proceed by induction . Assume positive definite . For x1 = 0 , ... , x , –1 = O we get q ( x ) = 9 ,,, 2 and hence q ,, > 0. Under this hypothesis we saw that q may be ...
... positive definite . The converse is trivial when r = 1 and we proceed by induction . Assume positive definite . For x1 = 0 , ... , x , –1 = O we get q ( x ) = 9 ,,, 2 and hence q ,, > 0. Under this hypothesis we saw that q may be ...
Page 338
... positive operator T with || T || ≤ 1. If the constant function 1 belongs to L and T is a positive operator such that T1 = 1 , then T is called a transition operator . ( It is automatically a contraction . ) Given two endomorphisms S ...
... positive operator T with || T || ≤ 1. If the constant function 1 belongs to L and T is a positive operator such that T1 = 1 , then T is called a transition operator . ( It is automatically a contraction . ) Given two endomorphisms S ...
Page 380
... positive and negative parts . Considering the first step of the random walk one sees that P { 1 > x } > P { X1 > x } and hence E ( X1 + ) ≤ E ( H1 ) < ∞ [ see V , ( 6.3 ) ] . The fact that is proper implies that with probability one ...
... positive and negative parts . Considering the first step of the random walk one sees that P { 1 > x } > P { X1 > x } and hence E ( X1 + ) ≤ E ( H1 ) < ∞ [ see V , ( 6.3 ) ] . The fact that is proper implies that with probability one ...
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