The combination of continuum many-body quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the cent...

Let 77l5..., 77w,... be a sequence of Independent integer-valued random variables. Let SnT]l + -~ + 7jn,Ari = ESn, B* = varS„,P„(m) = P(Sn = m), and f(t,rfj) denote the characteristic function of the random variable 77 •. The local limit theorem (LLT) is formulated as Pn(m) = (27rBy -exp{-(m^f /2B} + o(B~) when n-^00 uniformly for m. The first results on the normal approximation of binomial dis...

This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations1 applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a sequence of integral random variables {Ωn}n≥1 each of maximal span 1 (see below for definition), we a...

A well-known theorem of Lax and Wendroff states that if the sequence of approximate solutions to a system of hyperbolic conservation laws generated by a conservative consistent numerical scheme converges boundedly a.e. as the mesh parameter goes to zero, then the limit is a weak solution of the system. Moreover, if the scheme satisfies a discrete entropy inequality as well, the limit is an entr...

The de Moivre–Laplace theorem is a special case of the central limit for Bernoulli random variables, and can be proved by direct computation. We deduce any variable with finite variance from theorem. Our proof does not use advanced notions such as characteristic functions, Brownian motion, or stopping times.

Markov partitions work most efficiently for Anosov systems or for Axiom A systems. However, for hyperbolic dynamical systems which are either singular or whose hyperbolicity is nonuniform, the construction of a Markov partition, which in these cases is necessarily countable, is a rather delicate issue even when such a construction exists. An additional problem is the use of a countable Markov p...

We introduce a natural class of cellular automata characterised by a property of the local transition law without any assumption on the states set. We investigate some algebraic properties of the class and show that it contains intrinsically universal cellular automata. In addition we show that Rice’s theorem for limit sets is no longer true for that class, although infinitely many properties o...

We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the ‘spectrum’ of partitions lwn2N (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like ffiffiffiffiffiffiffiffiffiffi log n p , the corresponding random process in the bulk is shown to converge, in the s...

Abstract. We prove a limit theorem for an integral functional of a Markov process. The Markovian dynamics is characterized by a linear Boltzmann equation modeling a one-dimensional test particle of mass λ ≫ 1 in an external periodic potential and undergoing collisions with a background gas of particles with mass one. The object of our limit theorem is the time integral of the force exerted on t...

In this note, we present new asymptotic estimates comparing the word length and geodesic of closed geodesics on surfaces with (variable) negative sectional curvatures. particular, provide an averaged comparison these two important quantities obtain precise statistical results, including a central limit theorem local theorem. Further, as corollary also improve formula Sharp second author (1998)....