arXiv subjects
Publications and source records attributed to Kevin Garbe.
We explore the question whether Lipschitz functions of random variables under various forms of negative correlation satisfy concentration bounds similar to McDiarmid's inequality for independent random variables. We prove such a concentration bound for random variables satisfying the condition of negative regression, correcting an earlier proof by Dubhashi and Ranjan.
We examine the behavior of the coefficients of powers of polynomials over a finite field of prime order. Extending the work of Allouche-Berthe, 1997, we study a(n), the number of occurring strings of length n among coefficients of any power of a polynomial f reduced modulo a prime p. The sequence of line complexity a(n) is p-regular in the sense of Allouche-Shalit. For f=1+x and general p, we derive a recursion relation for a(n) then find a new formula for the generating function for a(n). We use the generating function to compute the asymptotics of a(n)/n^2 as n approaches infinity, which is an explicitly computable piecewise quadratic in x with n= [p^m/x] and x is a real number between 1/p and 1. Analyzing other cases, we form a conjecture about the generating function for general a(n). We examine the matrix B associated with f and p used to compute the count of a coefficient, which applies to the theory of linear cellular automata and fractals. For p=2 and polynomials of small degree we compute the largest positive eigenvalue, \lambda, of B, related to the fractal dimension d of the corresponding fractal by d= \log_2(\lambda). We find proofs and make a number of conjectures for some bounds on \lambda, and upper bounds on its degree.