M. Mandjes. Rare event of the state frequencies of a large number of Markov chains. To appear in: Stochastic Models, 1999. 19  Home/Search   Document Not in Database   Summary   Related Articles

....(12) where at the optimizing u it holds that the distribution f(u) is such that P d i=1 r i f i (u) c. The trajectory on the interval [0, u] can be found analogously to the three step recipe presented above (conditioning, large deviations on the individual terms, and Laplace ) see Mandjes [21]. After calculations we find f j (s) d X i,k=1 x i p ij (s)p jk (u s)y k , s # [0, u] 18) where x and y are such that d X k=1 x i p ik (u)y k = f i,0 and d X i=1 x i p ik (u)y k = f k (u) 19) As was extensively treated in [21] the resulting decay rate is d X i=1 (f ....

.... individual terms, and Laplace ) see Mandjes [21] After calculations we find f j (s) d X i,k=1 x i p ij (s)p jk (u s)y k , s # [0, u] 18) where x and y are such that d X k=1 x i p ik (u)y k = f i,0 and d X i=1 x i p ik (u)y k = f k (u) 19) As was extensively treated in [21], the resulting decay rate is d X i=1 (f 0,i log x i f i (u) log y i ) 20) where it is emphasized that x and y depend on u, as they are determined by (19) Finally consider (8) Based on (12) the path on [u, t] is heuristically derived analogously to (6) and is given by f j (s) d ....

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M. Mandjes. Rare event of the state frequencies of a large number of Markov chains. To appear in: Stochastic Models, 1999. 19

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