Some Research On Metric Properties And Convergence Theorem Of Random Continued Fractions

Given a list of random variables {An:n?1} defined above(?,F,P)and taken from natural numbers,define a random continuous fraction(?)Induced measures P(?)X-1 can contain all constant measures of the continuous fractional dynamic system on the unit interval through different distributions{An:n?1}.So the research on the random continued fractions is very important for that of corresponding dynamical systems.This paper focuses on the convergents and metric properties of random continuous fractions.Firstly,we discuss the basic properties of the numerator Pn(?)and the denominator QN(?)of its convergents(?)It is proved that the convergents almost surely converged to X(?),and the convergent speed of corresponding Levy constant of Qn(?)is further studied,that is,to estimate the probability(?)where ?>0.Fang et al(2015)proved that the above probability always has exponential upbound if {An:n?1} is ?-mixing.But the lower bound is missing.When the sequence{An:n?1}is independent and identical distributed(i.i.d.),the exponential lower bound estimate of the corresponding large deviation is given,also the exact values are obtained for some examples including Poission and geometric distributions.Then we indicate the average metric properties of non-negative multivariate functions f(Ai+1,Ai+2,…,Ai+k),and the order of the error term is obtained.This generalizes the results of Philipp(1967).Some applications are also given at the end.