Large Deviations for Partially Homogeneous Markov Chains
in Positive Quadrant

Mogulskii, Anatolii (Novosibirsk, Russia)
mogul@math.nsc.ru

(joint work with Akeksandr Borovkov)

In the talk so-called N -partially homogeneous (in space) Markov chains $X(y,n),~ n=0,1,2,...,~X(y,0)=y$, in positive quadrant ${ \mathbb R}^{2+}=\{x=(x_2,x_2): x_1\ge 0, x_2\ge 0\}$, are considered. These chains are characterized by the property of the transition probability

$P(y,A)={\mathsf P}(X(y,1)\in A)$:

• for some $N\ge 0$ the measure P(y,dx) in the domain $x_1\gt N, y_1\gt N$ depends on x₂, y₂, x₁-y₁ only, and in the domain $x_2\gt N, y_2\gt N$ -- on x₁, y₁, x₂-y₂ only.

For such chains the so-called integro-local large deviations principle has been obtained:

$$\lim_{n\to \infty} \frac{1}{\vert x\vert}\ln{\mathsf P}\Bigl(X(y,n)\in \Delta(x)\Bigr)= -D(\gamma,\beta,T),$$

where $\Delta(x)=[x_1,x_1+\Delta)\times [x_2,x_2+\Delta)$, $x\sim \beta \vert x\vert$, $y\sim \gamma \vert x\vert$, $n\sim T\vert x\vert$ and the rate function $D(\gamma,\beta,T)$ is known in explicit form. For ergodic Markov chains the precise asymptotic of the probabilities ${\mathsf P}(X(y,n)\in \Delta(x))$ have been found.