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

Random Continued fractions: L\'evy constant and Chernoff-type estimate