Central Limit Theorem and Diophantine Approximations

Invariance principles for Diophantine approximation of formal Laurent series over a finite base field

In a recent paper, the first and third author proved a central limit theorem for the number of coprime solutions of the diophantine approximation problem for formal Laurent series in the setting of the classical theorem of Khintchine. In this note, we consider a more general setting and show that even an invariance principle holds, thereby improving upon earlier work of the second author. Our r...

Central and Local Limit Theories in Markov Dependent Random Variables

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...

The Euclid algorithm is “totally” gaussian

We consider Euclid’s gcd algorithm for two integers (p, q) with 1 ≤ p ≤ q ≤ N , with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function d on the set of possible digits, asymptotically for N → ∞. For any additive cost of moderate growth d, Baladi and Vallée obtained a central limit theorem, and –in the ...

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.