Laws of the single logarithm for delayed sums of random fields

We extend a law of the single logarithm for delayed sums by Lai to delayed sums of random fields. A law for subsequences, which also includes the one-dimensional case, is obtained in passing.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ103 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Convergence rates for record times and the associated counting process

AbstractLet X1, X2,… be independent random variables with a common continuous distribution function. Rates of convergence in limit theorems for record times and the associated counting process are established. The proofs are based on inversion, a representation due to Williams and random walk methods

Strong limit theorems for random fields

The aim of the present paper is to review some joint work with Ulrich Stadtmüller concerning random field analogs of the classical strong laws. In the first half we start, as background information, by quoting the law of large numbers and the law of the iterated logarithm for random sequences as well as for random fields, and the law of the single logarithm for sequences. We close with a one-dimensional LSL pertaining to windows, whose edges expand in an “almost linear fashion”, viz., the length of the nth window equals, for example, n/ log n or n/ log log n. A sketch of the proof will also be given. The second part contains some extensions of the LSL to random fields, after which we turn to convergence rates in the law of large numbers. Departing from the now legendary Baum–Katz theorem in 1965, we review a number of results in the multiindex setting. Throughout main emphasis is on the case of “non-equal expansion rates”, viz., the case when the edges along the different directions expand at different rates. Some results when the power weights are replaced by almost exponential weights are also given. We close with some remarks on martingales and the strong law

The pedunculopontine tegmental nucleus - A functional hypothesis from the comparative literature

We present data from animal studies showing that the pedunculopontine tegmental nucleus-conserved through evolution, compartmentalized, and with a complex pattern of inputs and outputs-has functions that involve formation and updates of action-outcome associations, attention, and rapid decision making. This is in contrast to previous hypotheses about pedunculopontine function, which has served as a basis for clinical interest in the pedunculopontine in movement disorders. Current animal literature points to it being neither a specifically motor structure nor a master switch for sleep regulation. The pedunculopontine is connected to basal ganglia circuitry but also has primary sensory input across modalities and descending connections to pontomedullary, cerebellar, and spinal motor and autonomic control systems. Functional and anatomical studies in animals suggest strongly that, in addition to the pedunculopontine being an input and output station for the basal ganglia and key regulator of thalamic (and consequently cortical) activity, an additional major function is participation in the generation of actions on the basis of a first-pass analysis of incoming sensory data. Such a function-rapid decision making-has very high adaptive value for any vertebrate. We argue that in developing clinical strategies for treating basal ganglia disorders, it is necessary to take an account of the normal functions of the pedunculopontine. We believe that it is possible to use our hypothesis to explain why pedunculopontine deep brain stimulation used clinically has had variable outcomes in the treatment of parkinsonism motor symptoms and effects on cognitive processing. © 2016 International Parkinson and Movement Disorder Society