Given a sequence of complex square matrices, @math , consider the sequence of their partial products, defined by @math . What can be said about the asymptotics as @math of the sequence @math , where @math is a continuous function? A special case of our most general result addresses this question under the assumption that the matrices @math are an @math perturbation of a sequence of matrices with bounded partial products. We apply our theory to investigate the asymptotics of the approximants of continued fractions. In particular, when a continued fraction is @math limit 1-periodic of elliptic or loxodromic type, we show that its sequence of approximants tends to a circle in @math , or to a finite set of points lying on a circle. Our main theorem on such continued fractions unifies the treatment of the loxodromic and elliptic cases, which are convergent and divergent, respectively. When an approximating sequence tends to a circle, we obtain statistical information about the limiting distribution of the approximants. When the circle is the real line, the points are shown to have a Cauchy distribution with parameters given in terms of modifications of the original continued fraction. As an example of the general theory, a detailed study of a @math -continued fraction in five complex variables is provided. The most general theorem in the paper holds in the context of Banach algebras. The theory is also applied to @math -matrix continued fractions and recurrence sequences of Poincar 'e type and compared with closely related literature.
We are aware of four other places where work related to the results of this section was given previously. Two of these were motivated by the identity of Ramanujan. The first paper is @cite_46 which gave the first proof of . The proof in @cite_46 is particular to the continued fraction . However, section 3 of @cite_46 studied the recurrence @math and showed that when @math , the sequence @math has six limit points and that moreover a continued fraction whose convergents satisfies this recurrence under the @math assumption tends to three limit points (Theorem 3.3 of @cite_46 ). The paper does not consider other numbers of limits, however. Moreover, the role of the sixth roots of unity in the recurrence is not revealed. In the section 6 of the present paper, we treat the general case in which recurrences can have a finite or uncountable number of limits. Previously in @cite_20 we treated such recurrences with a finite number @math of limits as well as the associated continued fractions.
@cite_46 @cite_20
2036434814 2000931246 2073569022 2026924428
Abstract On page 45 in his lost notebook, Ramanujan asserts that a certain q -continued fraction has three limit points. More precisely, if A n B n denotes its n th partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of A n B n exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 For integers m⩾2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern–Stolz theorem. We give a theorem on a class of Poincare-type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity. We also generalize a curious q-continued fraction of Ramanujan's with three limits to a continued fraction with k distinct limit points, k⩾2. The k limits are evaluated in terms of ratios of certain q-series. Finally, we show how to use Daniel Bernoulli's continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m⩾2. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail.