Random continued fractions: Lévy constant and Chernoff-type estimate

Given a stochastic process {An,n≥1} taking values in natural numbers, the random continued fractions are defined as [A1,A2,⋯,An,⋯] analogously to the continued fraction expansion of real numbers. Assume that {An,n≥1} is ergodic and the expectation E(log⁡A1)<∞, we give a Lévy-type metric theorem which covers that of real case presented by Lévy in 1929. Moreover, a corresponding Chernoff-type estimate is obtained under the conditions that {An,n≥1} is ψ-mixing and for each 0