Shlomo Levental

A uniform CLT for uniformly bounded families of martingale differences

This paper develops a method to get empirical central limit theorems for martingale differences that are uniformly bounded. The main idea is to generalize to martingales some ideas of E. Gine and J. Zinn [Ann. Prob.12, 929–989 (1984)]. We consider two examples: An extension of a theorem of R. Dudley from i.i.d. to a certain type of Markov chain, and a uniform CLT for the “bakers transformation”.

Uniform CLT for Markov chains and its invariance principle: A martingale approach

The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.(7) Then we use the chaining argument with stratification adapted from that of Ossiander.(11) The results of this paper generalize those of Levental(10) and Ossiander.(11)

Uniform limit theorems for Harris recurrent Markov chains

SummaryWe study uniform limit theorems for regenerative processes and get strong law of large numbers and central limit theorem of this type. Then we apply those results to Harris recurrent Markov chains based on some ideas of K. Athreya, P. Ney and E. Nummelin.

THE UNIFORM CLT FOR MARTINGALE DIFFERENCE ARRAYS UNDER THE UNIFORMLY INTEGRABLE ENTROPY

In this paper we consider the uniform central limit theorem for a martingale-diere nce array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-diere nce arrays of process indexed by a class of measurable functions by a method as Ziegler (19) did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-diere nce and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler (19) and other results of independent problems. The results also generalizes those of Bae and Choi (3) to martingale-diere nce array of a function-indexed stochastic process. Finally, an application to classes of functions changing with n is given.

A Maximal Inequality for Real Numbers with Application to Exchangeable Random Variables

Let

Permutations, signs and the Brownian bridge

x=(x_1,\ldots, x_n)

Integrated Contract and Spot Market Procurement by a Risk-Averse Buying Firm

be a sequence of real numbers with

On almost sure convergence of the quadratic variation of Brownian motion

{\sum_{i=1}^n} x_i=0

Uniform CLT for Markov chains with a countable state space

, and let

Linked Recursive Preferences and Optimality

\Theta=\{\theta=(\theta_1,\ldots,\theta_n):\,\theta_i=\pm 1\}