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...

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela. Limit groups are finitely presented group...

We present a proof of a martingale central limit theorem (Theorem 2) due to McLeish (1974). Then, an application to Markov chains is given.

Now we recall the statement of the central limit theorem (CLT) and give a proof in the case of IID (independent identically distributed) random variables. The weak law of large numbers says that if Xn is a sequence of IID random variables with E[Xn] = 0, then writing Sn = ∑n−1 k=0 Xk, the time averages 1 n Sn converge to 0 in probability, or equivalently (since the limit is a constant), in dist...

One of the most useful generalizations of the central limit theorem is the martingale central limit theorem of Paul Lévy. Lévy was in part inspired by Lindeberg’s treatment of the central limit theorem for sums of independent – but not necessarily identically distributed – random variables. Lindeberg formulated what, in retrospect, is the right hypothesis, now known as the Lindeberg condition,1...

We characterize convergence of a sequence of d-dimensional random vectors by convergence of the one-dimensional margins and of the copula. The result is applied to the approximation of portfolios modelled by t-copulas with large degrees of freedom, and to the convergence of certain dependence measures of bivariate distributions. AMS 2000 Subject Classifications: primary: 60F05, 62H05 secondary:...

Thètrue' self-avoiding walk with bond repulsion is a nearest neighbour random walk on Z, for which the probability of jumping along a bond of the lattice is proportional to exp(?g number of previous jumps along that bond). First we prove a limit theorem for the distribution of the local time process of this walk. Using this result, later we prove a local limit theorem, as A ! 1, for the distrib...