A countable Markov chain in a Markovian environment is considered.A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly achieved.In order to prove this theorem,the entropy function h is i...A countable Markov chain in a Markovian environment is considered.A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly achieved.In order to prove this theorem,the entropy function h is introduced and the Shannon-McMillan-Breiman theorem for the Markov chain in a Markovian environment is shown. It's well-known that a Markov process in a Markovian environment is generally not a standard Markov chain,so an example of Poisson approximation for a process which is not a Markov process is given.On the other hand,when the environmental process degenerates to a constant sequence,a Poisson limit theorem for countable Markov chains,which is the generalization of Pitskel's result for finite Markov chains is obtained.展开更多
A Markovian risk process is considered in this paper, which is the generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a mode...A Markovian risk process is considered in this paper, which is the generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a model, the occurrence of claims is described by a point process {N（t）}t≥0 with N（t） being the number of jumps during the interval （0, t] for a Markov jump process. The ruin probability ψ（u） of a company facing such a risk model is mainly studied. An integral equation satisfied by the ruin probability function ψ（u） is obtained and the bounds for the convergence rate of the ruin probability ψ（u） are given by using a generalized renewal technique developed in the paper.展开更多
文摘A countable Markov chain in a Markovian environment is considered.A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly achieved.In order to prove this theorem,the entropy function h is introduced and the Shannon-McMillan-Breiman theorem for the Markov chain in a Markovian environment is shown. It's well-known that a Markov process in a Markovian environment is generally not a standard Markov chain,so an example of Poisson approximation for a process which is not a Markov process is given.On the other hand,when the environmental process degenerates to a constant sequence,a Poisson limit theorem for countable Markov chains,which is the generalization of Pitskel's result for finite Markov chains is obtained.
文摘A Markovian risk process is considered in this paper, which is the generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a model, the occurrence of claims is described by a point process {N（t）}t≥0 with N（t） being the number of jumps during the interval （0, t] for a Markov jump process. The ruin probability ψ（u） of a company facing such a risk model is mainly studied. An integral equation satisfied by the ruin probability function ψ（u） is obtained and the bounds for the convergence rate of the ruin probability ψ（u） are given by using a generalized renewal technique developed in the paper.
