Random Continued fractions: Lévy constant and Chernoff-type estimate

Abstract: Given a stochastic process ${A_n, n \geq 1}$ taking values in natural numbers, the random continued fractions is defined as $[A_1, A_2, \cdots, A_n, \cdots]$ analogue to the continued fraction expansion of real numbers. Assume that ${A_n, n \geq 1}$ is ergodic and the expectation $E(\log A_1) &lt; \infty$, we give a L\'evy-type metric theorem which covers that of real case presented by L\'evy in 1929. Moreover, a corresponding Chernoff-type estimate is obtained under the conditions ${A_n, n \geq 1}$ is $\psi$-mixing and for each $0< t< 1$, $E(A_1^t) &lt; \infty$.