CONTINUED FRACTIONS WITH ODD PARTIAL QUOTIENTS

Every Euclidean algorithm is associated with a kind of continued fraction representation of a number. The representation associated with "odd" Euclidean algorithm we will call "odd" continued fraction. We consider the limit distribution function F(x) for sequences of rationals with bounded sum of partial quotients for "odd" continued fractions. In this paper we prove certain properties of the function F(x). Particularly this function is singular and satisfies a number of functional equations. We also show that the value F(x) can be expressed in terms of partial quotients of the "odd" continued fraction representation of a number x.

Longest Increasing Subsequences of Random Colored Permutations

We compute the limit distribution for the (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In the two–colored case our method provides a different proof of a similar result by Tracy and Widom about the longest increasing subsequences of signed permutations (math.CO/9811154). Our main idea is to reduce the 'colored' problem to the case of usual random permutations using certain combinatorial results and elementary probabilistic arguments.

On the convergence of the supercritical branching processes with immigration

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

On the convergence of the supercritical branching processes with immigration

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn } exists such that {Xn/cn } converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 &lt; m &lt; ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn } such that {Xn /c n} converges in law to a proper limit distribution function F, with F(0 +) &lt; 1.