A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs

A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs

For large N , we consider the ordinary continued fraction of x = p/q with 1 ≤ p ≤ q ≤ N , or, equivalently, Euclid’s gcd algorithm for two integers 1 ≤ p ≤ q ≤ N , putting the uniform distribution on the set of p and qs. We study the distribution of the total cost of execution of the algorithm for an additive cost function c on the set Z∗+ of possible digits, asymptotically for N → ∞. If c is n...

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

Central Limit Theorem and Diophantine Approximations

Let Fn denote the distribution function of the normalized sum Zn = (X1+ · · ·+Xn)/σ √ n of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of Fn to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of Fn by the Edgeworth corrections (modulo logarithmically growing factors in n) are given in ter...

Convergence rates for the central limit theorem.

Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars

A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars (MSCs) where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges...