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Given a sequence of complex square matrices, $a_n$, consider the sequence oftheir partial products, defined by $p_n=p_{n-1}a_{n}$. What can be said aboutthe asymptotics as $n\to\infty$ of the sequence $f(p_n)$, where $f$ is acontinuous function? A special case of our most general result addresses thisquestion under the assumption that the matrices $a_n$ are an $l_1$ perturbationof a sequence of matrices with bounded partial products. We apply our theory toinvestigate the asymptotics of the approximants of continued fractions. Inparticular, when a continued fraction is $l_1$ limit 1-periodic of elliptic orloxodromic type, we show that its sequence of approximants tends to a circle in$\hat{\mathbb{C}}$, or to a finite set of points lying on a circle. Our maintheorem on such continued fractions unifies the treatment of the loxodromic andelliptic cases, which are convergent and divergent, respectively. When anapproximating sequence tends to a circle, we obtain statistical informationabout the limiting distribution of the approximants. When the circle is thereal line, the points are shown to have a Cauchy distribution with parametersgiven in terms of modifications of the original continued fraction. As anexample of the general theory, a detailed study of a $q$-continued fraction infive complex variables is provided. The most general theorem in the paper holdsin the context of Banach algebras. The theory is also applied to $(r,s)$-matrixcontinued fractions and recurrence sequences of Poincar\'e type and comparedwith closely related literature.

Asymptotics and Sequential Closures of Continued Fractions and Generalizations