Approximating Countable Markov Chains |
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Page 32
... Prob A = IIM = 1 ( Prob Am ) ( Prob Bm ) . · · n ( m ) — n ( m1 ) and d ( m ) = Σƒ d ( f ) = n ( m ) — n ( m − 1 ) and s ( m ) = tm - tm - 1 . Use ( 69b ) and the fact that Z1 , Z2 , ... are inde- pendent with common distribution p : Prob ...
... Prob A = IIM = 1 ( Prob Am ) ( Prob Bm ) . · · n ( m ) — n ( m1 ) and d ( m ) = Σƒ d ( f ) = n ( m ) — n ( m − 1 ) and s ( m ) = tm - tm - 1 . Use ( 69b ) and the fact that Z1 , Z2 , ... are inde- pendent with common distribution p : Prob ...
Page 47
... Prob { U¡ ≤ s ≤ t < U ; + U ; } e ' -ct = Q ( i , j ) · e ̄α . = • S S ebu du = Q¿ ( i , j ) · [ s + o ( t ) ] as t → 0 . i Remember that ƒ is the least t with X ( t ) = j . The main thing to see is ( 99 ) If Q¡ ( i , i ) > 0 and 0 ...
... Prob { U¡ ≤ s ≤ t < U ; + U ; } e ' -ct = Q ( i , j ) · e ̄α . = • S S ebu du = Q¿ ( i , j ) · [ s + o ( t ) ] as t → 0 . i Remember that ƒ is the least t with X ( t ) = j . The main thing to see is ( 99 ) If Q¡ ( i , i ) > 0 and 0 ...
Page 50
... Prob { G ( T− ) ≤ ( 1 + a ) T and T≤t < T + S } . And P¡ { A , ( t ) } = á ( i , j ) q , where q = Prob { T ≤ t < T + S } . Let m be the joint distribution of T and S , a probability on the positive quadrant . By Fubini , р = S 0 ...
... Prob { G ( T− ) ≤ ( 1 + a ) T and T≤t < T + S } . And P¡ { A , ( t ) } = á ( i , j ) q , where q = Prob { T ≤ t < T + S } . Let m be the joint distribution of T and S , a probability on the positive quadrant . By Fubini , р = S 0 ...
Contents
RESTRICTING THE RANGE | 1 |
RESTRICTING THE RANGE APPLICATIONS | 64 |
CONSTRUCTING THE GENERAL MARKOV CHAIN | 95 |
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1)-intervals a₁ absorbing argue argument b₂ binary rationals Brownian Motion chapter coincides conditional distribution construction converges David Freedman defined exponentially distributed F(1)-measurable Figure finite subset Fubini hitting holding i₁ implies In+1 independent and exponential infinite interval of constancy j₁ joint distribution jump Lebesgue measure Lemma Let f locally finitary Markov chain Markov process Markov property Markov with stationary Markov with transitions Math nondecreasing notation null set P-distribution P-probability P₁ Poisson process Poisson with parameter positive Prob probability triple product measurable prove pseudo-jumps Qn(i QN(j Qn+1 quasiregular random variables recurrent restriction retracted right continuous sample functions satisfies Section sequence spends interior standard stochastic semigroup starting stationary standard transitions stationary transitions strictly increasing Suppose T₁ Theorem TJ,n TJ,o TN,o visits VOLKER STRASSEN WILLIAM FELLER X₁ XN+1 YN+m