Elements of the Theory of Markov Processes and Their ApplicationsGraduate-level text and reference in probability, with numerous scientific applications. Nonmeasure-theoretic introduction to theory of Markov processes and to mathematical models based on the theory. Appendixes. Bibliographies. 1960 edition. |
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Page 25
... Hence , F ' ( 1 ) exists and is < 1 . Conversely , if F ′ ( 1 ) ≤ 1 , F ' ( s ) is either constant in [ 0,1 ) or strictly increasing with s . Hence , F ' ( s ) < 1 in either case . Application of the mean - value theorem gives F ( s ) ...
... Hence , F ' ( 1 ) exists and is < 1 . Conversely , if F ′ ( 1 ) ≤ 1 , F ' ( s ) is either constant in [ 0,1 ) or strictly increasing with s . Hence , F ' ( s ) < 1 in either case . Application of the mean - value theorem gives F ( s ) ...
Page 29
... Hence , for every N < > 0 we can select an N such that Σ K N terms in the series p N n = 1 N = Σ Kp v = 1 = 1 e . Now , the last ( - ) differ arbitrarily little from 7 , K , and hence from 7 ,. Finally , the sum of the first n - Σ n = 1 ...
... Hence , for every N < > 0 we can select an N such that Σ K N terms in the series p N n = 1 N = Σ Kp v = 1 = 1 e . Now , the last ( - ) differ arbitrarily little from 7 , K , and hence from 7 ,. Finally , the sum of the first n - Σ n = 1 ...
Page 69
... hence , we can rewrite it as 00 F1 , ( t ) = dis + + S " ( 2 au P2 ( 7 ) dr F ij 0 k = 1 kj ( 2.66 ) Since Σ | a ... ] < ∞ and 0≤ F * ; ( 7 ) < 1 , the series in ( 2.66 ) is k = 1 uniformly convergent for all for all 00 > 0 . Hence ...
... hence , we can rewrite it as 00 F1 , ( t ) = dis + + S " ( 2 au P2 ( 7 ) dr F ij 0 k = 1 kj ( 2.66 ) Since Σ | a ... ] < ∞ and 0≤ F * ; ( 7 ) < 1 , the series in ( 2.66 ) is k = 1 uniformly convergent for all for all 00 > 0 . Hence ...
Other editions - View all
Elements of the Theory of Markov Processes and Their Applications A. T. Bharucha-Reid Limited preview - 2012 |
Elements of the Theory of Markov Processes and Their Applications Albert T. Bharucha-Reid Limited preview - 1997 |
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