In two recent works, Kuba and Mahmoud (2015a) and (2015b) introduced the family of two-color affine balanced Pólya urn schemes with multiple drawings. We show that, in large-index urns (urn index between ½ and 1) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new, even in the standard model when only one ball is drawn from the urn in each step (except for the classical Pólya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.

Let (W n (θ))n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_∞(θ) its limit. Assuming essentially that the martingale (W n (2θ))n∈ℕ0 is uniformly integrable and that var W 1(θ) is finite, we prove a functional central limit theorem for the tail process (W ∞(θ)-W n+r (θ))r∈ℕ0 and a law of the iterated logarithm for W ∞(θ)-W n (θ) as n→∞.

We study the asymptotic behaviors of estimators of the parameters in an Ornstein–Uhlenbeck process with linear drift, such as the law of the iterated logarithm (LIL) and Berry–Esseen bounds. As an application of the Berry–Esseen bounds, the precise rates in the LIL for the estimators are obtained.

The conditions in the strong law of large numbers given by Li et al. [‘A strong law for B-valued arrays’, Proc. Amer. Math. Soc.123 (1995), 3205–3212] for B-valued arrays are relaxed. Further, the compact logarithm rate law and the bounded logarithm rate law are discussed for the moving average process based on an array of random elements.

In [6], Walter Philipp wrote that “… the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.” Many authors [1], [2], [4], [5], [9] have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables.

We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn [8] for certain Gibbs measures. These do not fall into the $\phi $-mixing or strong mixing cases established in the literature, but are needed for our investigations [7] into diffusions on configuration space.

The proofs are all obtained by patching together standard results from [5], [9] while keeping a careful eye on the correlations.

The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

In this paper we connect various topological and probabilistic forms of stability for discrete-time Markov chains. These include tightness on the one hand and Harris recurrence and ergodicity on the other. We show that these concepts of stability are largely equivalent for a major class of chains (chains with continuous components), or if the state space has a sufficiently rich class of appropriate sets (‘petite sets').

We use a discrete formulation of Dynkin's formula to establish unified criteria for these stability concepts, through bounding of moments of first entrance times to petite sets. This gives a generalization of Lyapunov–Foster criteria for the various stability conditions to hold. Under these criteria, ergodic theorems are shown to be valid even in the non-irreducible case. These results allow a more general test function approach for determining rates of convergence of the underlying distributions of a Markov chain, and provide strong mixing results and new versions of the central limit theorem and the law of the iterated logarithm.

We show that a theory for stopped two-dimensional random walks is well suited to describe cumulative shock models. Limit theorems for the lifetime/failure time of a system are provided.

Traditional methods for detecting the hidden frequencies in white or coloured noise is based on the distribution of periodogram ordinates Ι (2πj/N) and hypothesis testing. The power is low when a hidden frequency falls midway between two 2nj/N and it is difficult to discuss the consistency of the procedures. Using theorems on almost sure convergence of the periodogram, this paper offers a procedure for detecting and estimating the hidden frequencies, and discusses the consistency and the rates of convergence. Simulation shows very good results, when the spectrum of the noise is not too steep.

This paper considers the problem of estimating the growth rate ρ of a p-type Galton–Watson process {Zn}. To this end, a general approach of possible independent interest to central limit theorems for discrete-time branching processes is developed. The idea is to adapt martingale central limit theory to martingale difference triangular arrays indexed by the set of all individuals ever alive. Iterated logarithm laws are derived by similar methods. Asymptotic distribution results and the a.s. asymptotic behaviour are derived for a maximum likelihood estimator based upon all parent–offspring combinations in a given number N of generations, and for the estimator which depends on the total generation sizes only.

A model is considered for a situation in which one rumour suppresses another in a closed population. The distribution of the maximum value attained by the proportion spreading the weaker rumour is obtained in the asymptotic case, and this is compared with some actual distributions for finite population size. Closer approximations to the latter distributions are obtained.

Strassen-type law of the iterated logarithm for Brownian sheets presented by Pyke [7] is proved by using recent results of Kuelbs and Lepage [4]: the law of the iterated logarithm for Brownian motion in a Banach space and some applications are given.

The familiar three theorems of Rényi concerning the record times in an i.i.d. sequence of random variables are extended to the record times and inter-record times of a sequence of dependent, non-identically distributed random variables defined on a finite Markov chain. These theorems are the Central Limit Theorem (C.L.T.), the Strong Law of Large Numbers (S.L.L.N.) and the Law of the Iterated Logarithm (L.I.L.). Similar results are also obtained for m-record times, inter-m-record times, and for the continuous parameter situation when observations are taken at the epochs of a Poisson process.