Showing papers by "Richard Arratia published in 1993"

On random polynomials over finite fields

Abstract: We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.

BOOK REVIEW D. Arxmus, Probability Approximations via the Poisson Clumping Heuristic. Springer, Berlin, 1989.

Abstract: It might be that the two fundamental distributions in probability are the normal and the Poisson. Making these into processes with independent incre-Nevertheless, graduate training in probability tends to ignore Poisson processes and none of the almost 200 entries in the Probability and Statistics category of the 1991 Mathematics Subject Classification explicitly mentions Poisson, either. Brownian motion is a fascinating object in its own right, even without considering its relation to martingales, diffusions, stochastic differential equations, and so on. In contrast, it is only the comparison of Poisson processes to various dependent processes that makes for rewarding study. Two recent books, Probability Approximations via the Poisson Clumping Heuristic (PCH) and Poisson Approximation (PA), combine to reveal the enormous depth and complexity of applications of the Poisson distribution and Poisson processes. These books are complementary. In the language of Breiman (19681, PCH addresses the left hand, intuition, and PA addresses the right hand, technique. Poisson approximations for a given probability model involve three ingredients. The first, most easily overlooked, is the identification of suitable things to count, so that the random number W of occurrences is approximately Poisson. The second is an evaluation of, or approximation for, the Poisson parameter A = EW. The third ingredient is an analysis of the dependence structure to show that W is close to PdA), the Poisson distribution with parameter A. Informally, PCH discusses the first two ingredients, and PA is concerned with the third. is titled " Observations fitting the Poisson distribution " and its footnote states: The Poisson distribution has become known as the law of small numbers or of rare events. These are misnomers which proved detrimental to the realization of the fundamental role of the Poisson distribution. The following examples will show how misleading the two names are.