## Markov additive processes. I: Eigenvalue properties and limit theorems.(English)Zbl 0625.60027

Markov additive processes are of the form $$\{(X_ n,S_ n):n=0,1,2,...\}$$ where $$X_ n$$ is a Markov chain on $${\mathbb{E}}$$ (with $$\sigma$$-field $${\mathcal E})$$ and $$S_ n=\sum^{n}_{i=1}\xi_ i$$ $$(\in {\mathbb{R}}^ d)$$, and the distribution of $$(X_{n+1},\xi_{n+1})$$ depends on the past only through $$X_ n$$. Let $$I(\cdot)$$ be the indicator function and $$E_ x$$ denote the expectation conditional on $$X_ 0=x.$$
In this paper a careful study is made of the eigenvalues and eigenvectors of the kernel defined for $$(x,A)\in {\mathbb{E}}\times {\mathcal E}$$ by $$\hat P(\alpha) = E_ x(e^{\alpha \xi}I(X_ 1\in A))$$ for $$\alpha \in {\mathbb{R}}^ d$$. This leads rather naturally to large deviation results for the process. These give information on the large deviation probabilities $n^{-1} \log (E_ xI(X_ n\in A,\quad n^{-1} S_ n\in \Gamma))$ for large n and certain $$\Gamma$$. These results are obtained under the condition that a certain multidimensional Laplace transform has an open set as its domain of finiteness.
The sequel, part II, see the following title, Zbl 0625.60028, extends these results using a truncation argument and hence establishes that Markov additive processes obey a large deviation principle. In particular, the lower bound on the large deviation probabilities is obtained under very weak hypotheses.
Reviewer: J.D.Biggins

### MSC:

 60F10 Large deviations 60J05 Discrete-time Markov processes on general state spaces 60K15 Markov renewal processes, semi-Markov processes

### Keywords:

Markov additive processes; large deviation

Zbl 0625.60028
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