In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/∞. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods...

Current evolutionary biology models usually assume that a phenotype undergoes gradual change. This is in stark contrast to biological intuition which indicates that change can also be punctuated — the phenotype can jump. Such a jump can especially occur at speciation, i.e. dramatic change occurs that drives the species apart. Here we derive a central limit theorem for punctuated equilibrium. We...

The Edwards model in one dimension is a transformed path measure for standard Brownian motion discouraging self-intersections. We prove a central limit theorem for the endpoint of the path, extending a law of large numbers proved by Westwater (1984). The scaled variance is characterized in terms of the largest eigenvalue of a one-parameter family of differential operators, introduced and analyz...

The aim is to approve the Central limit theorem on MV-algebras by the new approach, presented by Riečan in [1]. We use the observable as a distribution function instead of the σ-homomorphism. The main idea is in local representation of σ-algebras. The following theorem is proved: Let M be a σ-complete MV-algebra with product, m : M −→ 〈0, 1〉 be a σadditive state, (xn)n be a sequence of independ...

Motivated by Barron (1986, Ann. Probab. 14, 336–342), Brown (1982, Statistics and Probability: Essays in Honour of C.R. Rao, pp. 141–148) and Carlen and So er (1991, Comm. Math. Phys. 140, 339–371), we prove a version of the Lindeberg–Feller Theorem, showing normal convergence of the normalised sum of independent, not necessarily identically distributed random variables, under standard conditio...

It has long been known that the standard Riemann integral, encountered first as one of the most important concepts of calculus, has many shortcomings, and can only be defined over the limited class of sets which have boundaries of Jordan content zero. In this paper I present the measure theory necessary to develop the more robust Lebesgue integral. This is interesting in its own right, and alth...

2 Poisson approximation 6 2.1 A coupling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Stein’s method for Poisson approximation . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Independent summands . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Dependent summands: the local approach . . . . . . . . . . . . . . . . . 10 2.2.3 Size biasing and coup...

A new variant of CLT is established for random elds de ned on Rd which are strictly stationary, with nite second moment and weakly dependent (comprising cases of positive or negative association). The summation domains grow in the van Hove sense. Simultaneously the indices of observations form more and more dense grids in these domains. Thus the e ect of combining two scaling procedures is stud...

A unified approach to obtaining the central limit theorem for hyperbolic dynamical systems is presented. It builds on previous results for one dimensional maps but it applies to the multidimensional case as well. CONTENT 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 2 1. A general probabilistic result . . . . ....

Often, our goal is to understand how collections of a large number of independent random variables behave. This is what the laws of large numbers reveal. In general, the idea is that the average of a large number of i.i.d. random variables will approach the expectation. Sometimes that is enough, but usually, we also need to understand how fast does the average converge to the expectation. We sa...