We will investigate the large deviation rates for sums of the form $\sum_i f(x_i) g(x_{2i})$ where $\{x_i\}$ is a nice Markov process. In other words calculate

$\lim_{n\to\infty}{1\over n}\log E[\exp \sum_{i=1}^n f(x_i) g(x_{2i})]$

where $\{x_i\}$ is Markov Chain with transition probability $\pi(x, y)$.