Let X; X i ; i2N, be independent identically distributed random variables. It is shown that the Student t--statistic based upon the sample fX i g n i=1 is asymptotically N(0;1) if and only if X is in the domain of attraction of the normal law. It is also shown that, for any X, if the self--normalized sums Un:= P n i=1 X i ffi\Gamma P n i=1 X 2 i \Delta 1=2 ; n2N, are stochastically bounded then they are uniformly subgaussian that is, sup n E exp(U 2 n )!1 for some ?0.

We define a local time flow of skew Brownian motions, that is, a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but driven by the same Brownian motion. We prove several results on distributional and path properties of the flow. Our main result is a version of the Ray–Knight theorem on local times. In our case, however, the local time process viewed as a function of the spatial variable is a pure jump Markov process rather than a diffusion.

We show that far from capturing a formally new phenomenon, informational herding is really a special case of single-person experimentation — and ‘bad herds’ the typical failure of complete learning. We then analyze the analogous team equilibrium, where individuals maximize the present discounted welfare of posterity. To do so, we generalize Gittins indices to our non-bandit learning problem, and thereby characterize when contrarian behaviour arises: (i) While herds are still constrained efficient, they arise for a strictly smaller belief set. (ii) A log-concave log-likelihood ratio density robustly ensures that individuals should lean more against their myopic preference for an action the more popular it becomes. We thank Patrick Bolton and three referees in guiding this radical revision, as well as Abhijit Banerjee, Katya Malinova, Meg Meyer, Christopher Wallace, and seminar participants at the MIT theory lunch, the

Motivated by models of queues with server vacations, we consider a Levy process modified to have random jumps at arbitrary stopping times. The extra jumps can counteract a drift in the Le vy ´ process so that the overall Levy process with secondary jump input, can have a proper limiting distribution. For example, the workload process in an M/G/1 queue with a server vacation each time the server finds an empty system is such a Le vy ´ process with secondary jump input. We show that a certain functional of a Le vy ´ process with secondary jump input is a martingale and we apply this martingale to characterize the steady-state distribution. We establish stochastic decomposition results for the case in which the Levy process has no negative jumps, which extend and unify previous decomposition results for the workload process in the M/G/1 queue with server vacations and Brownian motion with secondary jump input. We also apply martingales to provide a new proof of the known simple form of the steady-state distribution of the associated reflected Levy process when the Levy process has no negative jumps (the generalized Pollaczek-Khinchine formula).

After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its set-theoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness ” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measure-theoretical codification of stochastic processes genuine chance processes can be defined rigorously as so-called regular processes which do not allow a long-term prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes.

This thesis considers continuous time autoregressive processes defined by stochastic differential equations and develops some methods for modelling time series data by such processes. The first part of the thesis looks at continuous time linear autoregressive (CAR) processes defined by linear stochastic differential equations. These processes are well-understood and there is a large body of literature devoted to their study. I summarise some of the relevant material and develop some further results. In particular, I propose a new and very fast method of estimation using an approach analogous to the Yule–Walker estimates for discrete time autoregressive processes. The models so estimated may be used for preliminary analysis of the appropriate model structure and as a starting point for maximum likelihood estimation. A natural extension of CAR processes is the class of continuous time threshold autoregressive (CTAR) processes defined by piecewise linear stochastic differential

A vector autoregression is singular when explosive characteristic roots have geometric multiplicity larger than one. The singular component is a mixingale. Martingale decompositions are constructed for sample moments involving the singular component. This permits weak and strong analysis in the case of martingale difference innovations. While least squares estimators are shown to be inconsistent in the singular case, procedures for lag length determination are shown to have the same asymptotic properties in regular and singular cases.

We consider random walks in a random environment which are generalized versions of well-known effective models for Mott variablerange hopping. We study the homogenized diffusion constant of the random walk in the one-dimensional case. We prove various estimates on the low-temperature behavior which confirm and extend previous work by physicists. 1. Introduction. Random

Consider a random walk (Sn:n ≥ 0) with drift −µ and S0 = 0. Assuming that the increments have exponential moments, negative mean, and are strongly nonlattice, we provide a complete asymptotic expansion (in powers of µ> 0) that corrects the diffusion approximation of the all time maximum M = maxn≥0 Sn. Our results extend both the first-order correction of Siegmund [Adv. in Appl. Probab. 11 (1979) 701–719] and the full asymptotic expansion provided in the Gaussian case by Chang and Peres [Ann. Probab. 25 (1997) 787–802]. We also show that the Cramér–Lundberg constant (as a function of µ) admits an analytic extension throughout a neighborhood of the origin in the complex plane C. Finally, when the increments of the random walk have nonnegative mean µ, we show that the Laplace transform, Eµ exp(−bR(∞)), of the limiting overshoot, R(∞), can be analytically extended throughout a disc centered at the origin in C × C (jointly for both b and µ). In addition, when the distribution of the increments is continuous and appropriately symmetric, we show that EµSτ [where τ is the first (strict) ascending ladder epoch] can be analytically extended to a disc centered at the origin in C, generalizing the main result in [Ann. Probab. 25 (1997) 787–802] and extending a related result of Chang [Ann. Appl. Probab. 2 (1992) 714–738]. 1. Introduction. Let (Xn:n ≥ 1

This paper derives the exact distribution of the maximum likelihood estimator of a first order linear autoregression with an exponential disturbance term. We also show that even if the process is stationary, the estimator is T-consistent, where T is the sample size. In the unit root case the estimator is T²-consistent, while in the...