Exponential inequalities and the law of the iterated logarithm in the unbounded forecasting game

We study the law of the iterated logarithm in the framework of game-theoretic probability of Shafer and Vovk. We investigate hedges under which a game-theoretic version of the upper bound of the law of the iterated logarithm holds without any condition on Reality’s moves in the unbounded forecasting game. We prove that in the unbounded forecasting game with an exponential hedge, Skeptic can force the upper bound of the law of the iterated logarithm without conditions on Reality’s moves. We give two examples such a hedge. For proving these results we derive exponential inequalities in the game-theoretic framework which may be of independent interest. Finally, we give related results for measure-theoretic probability which improve the results of Liu and Watbled (Stochastic Processes and their Applications 119:3101–3132, 2009).     

On functional laws of the iterated logarithm

Summary A Skorokhod embedding approach is used to give functional laws of the iterated logarithm which involve the process up to timen in the reverse martingale case and the tail of the process in the martingale case. This complements the more usual versions of the iterated logarithm laws for martingales and reverse martingales.     

Strong laws of large numbers for sub-linear expectations

We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed(IID) random variables for sub-linear expectations initiated by Peng.It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.     

A note on some laws of the iterated logarithm

Some function space laws of the iterated logarithm for Brownian motion with values in finite and infinite dimensional vector spaces are shown to follow from Hincin's classical law of the iterated logarithm and some martingale techniques. A law of the iterated logarithm for Brownian motion in a differentible manifold is also stated.     

Some laws of the iterated logarithm in Hilbertian autoregressive models

We consider the law of the iterated logarithm for the empirical covariance of Hilbertian autoregressive processes. As an application, we obtain laws of the iterated logarithm for the eigenvalues and associated projectors of the empirical covariance.     

Laws of iterated logarithm for state variables with applications

Letx _k be the state variable (solution) of a stochastic difference equation. This paper gives the laws of iterated logarithm for {x _k} and {x _kx _k ^τ }, which being strongly correlated, are neither stationary nor ergodic. The results obtained are then applied to Kalman filter and LQG control problem. Work supported by National Natural Science Foundation of China and the TWAS Research Grant No. 87-43.     

Transformations in functional iterated logarithm laws and regular variation

Summary It is shown that functional iterated logarithm (log log) laws for geometric subsequences imply the corresponding laws for full sequences, and that the converse is not true. The implication is proved by simple algebraic arguments of regular variation type.Supported in part by a NATO grant for international collaboration in research     

A tail empirical process approach to some nonstandard laws of the iterated logarithm

We show that a number of nonstandard laws of the iterated logarithm have limiting constants which may be expressed as the extreme values of functionals off, wheref varies over suitable compact sets of functions. By solving the corresponding extremal problems, we show how these constants are generated. In addition, several new laws are presented.     

A connection between Strassen's and Donsker-Varadhan's laws of the iterated logarithm

Summary A connection is given between various functional laws of the iterated logarithm for Brownian motion due to Donsker and Varadhan, Csáki, Chung, Strassen, and the author.Supported by an NSF grant     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

Exponential estimate for the law of the iterated logarithm in banach space

Obninsk Institute of Atomic Energy. Translated from Matematicheskie Zametki, Vol. 56, No. 5, pp. 98–107, November, 1994.     

Small values of Gaussian processes and functional laws of the iterated logarithm

Summary We estimate small ball probabilities for locally nondeterministic Gaussian processes with stationary increments, a class of processes that includes the fractional Brownian motions. These estimates are used to prove Chung type laws of the iterated logarithm.Research supported by the United States Air Force office of Scientific Research, Contract No. 91-0030     

Bounded laws of the iterated logarithm for quadratic forms in Gaussian random variables

In this paper we prove bounded laws of the iterated logarithm for Gaussian quadratic forms. The underlying sequence of Gaussian variables is assumed to satisfy quite general conditions on its covariance structure. Basic tools are maximal inequalities of exponential type for sums of dependent random variables which may be of own interest. Several examples illustrate the sharpness of the results. In a particular section the bounded law of the iterated logarithm is shown for quadratic variation of Brownian motion.     

Functional law of the iterated logarithm for fields and its applications

For a Wiener field with an arbitrary finite number of parameters, we construct the law of the iterated logarithm in the functional form. We consider the problem for random fields of a certain type to reside within curvilinear boundaries without assuming that the Cairoli—Walsh condition is satisfied. Donetsk University, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 883–894, July, 1997.     

Generalized one-sided laws of the iterated logarithm for random variables barely with or without finite mean

The almost sure limiting behavior of weighted sums of independent and identically distributed random variables barely with or without finite mean are established. Results for these partial sums, $$\sum\limits_{k = 1}^n {k^\alpha X_k ,} \alpha \in R$$ have been studied, but only when α=?1 or α=0. As it turns out, the two cases of major interest are α=?1 and α>?1. The purpose of this article is to examine the latter.     

Complete moment convergence for ND random variables under the sub-linear expectations

In this article, we establish a general result on complete moment convergence for arrays of rowwise negatively dependent(ND) random variables under the sub-linear expectations. As applications, we can obtain a series of results on complete moment convergence for ND random variables under the sub-linear expectations.