Some Tauberian Remainder Theorems for Holder Summability

In this paper, we prove some Tauberian remainder theorems that generalize the results given by Meronen and Tammeraid [Math. Model. Anal., 18(1):97– 102, 2013] for Holder summability method using the notion of the general control modulo of the oscillatory behaviour of nonnegative integer order

Necessary and sufficient Tauberian conditions for the A^r method of summability

Móricz and Rhoades determined the necessary and sufficient Tauberian conditions for certain weighted mean methods of summability in [Acta. Math. Hungar. 102(4) (2004), 279{285]. In the present paper, we deal with the necessary and sufficient Tauberian conditions for the Ar method which was introduced by Bas̨ar in [Fırat Üniv. Fen & Müh. Bil. Dergisi 5(1)(1993), 113{117]

Sieve-based inference for infinite-variance linear processes

We extend the available asymptotic theory for autoregressive sieve estimators to cover the case of stationary and invertible linear processes driven by independent identically distributed (i.i.d.) infinite variance (IV) innovations. We show that the ordinary least squares sieve estimates, together with estimates of the impulse responses derived from these, obtained from an autoregression whose order is an increasing function of the sample size, are consistent and exhibit asymptotic properties analogous to those which obtain for a finite-order autoregressive process driven by i.i.d. IV errors. As these limit distributions cannot be directly employed for inference because they either may not exist or, where they do, depend on unknown parameters, a second contribution of the paper is to investigate the usefulness of bootstrap methods in this setting. Focusing on three sieve bootstraps: the wild and permutation bootstraps, and a hybrid of the two, we show that, in contrast to the case of finite variance innovations, the wild bootstrap requires an infeasible correction to be consistent, whereas the other two bootstrap schemes are shown to be consistent (the hybrid for symmetrically distributed innovations) under general conditions

Some one-sided conditions under which subsequential convergence follows from (A, k) summability method

WOS: 000287564000021In this paper, we introduce some conditions to recover subsequential convergence of an (A, k) summable sequence which is one-sidedly bounded by another sequence. (C) 2010 Elsevier Ltd. All rights reserved

Some one-sided conditions under which subsequential convergence follows from (A,k) summability method

In this paper, we introduce some conditions to recover subsequential convergence of an (A,k) summable sequence which is one-sidedly bounded by another sequence. © 2011 Elsevier Ltd. All rights reserved

Mean-field driven first-order phase transitions in systems with long-range interactions

We consider a class of spin systems on $\Z^d$ with vector valued spins (\bS_x) that interact via the pair-potentials J_{x,y} \bS_x\cdot\bS_y. The interactions are generally spread-out in the sense that the $J_{x,y}$'s exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions $d\ge3$, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions $d=1,2$ for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique ``state,'' then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure.Comment: 57 pages; uses a (modified) jstatphys class fil

Heat Kernels, Stochastic Processes and Functional Inequalities

The general topic of the 2013 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments: finite and infinite-dimensional manifolds, metric spaces, fractals and graphs, including random environments. The workshop brought together leading researchers from analysis, probability and geometry and provided a unique opportunity for interaction of established and young scientists from these areas. Unifying themes were heat kernel analysis, mass transport problems and related functional inequalities such as Poincar´e, Sobolev, logarithmic Sobolev, Bakry-Emery, Otto-Villani and Talagrand inequalities. These concepts were at the heart of Perelman’s proof of Poincar´e’s conjecture, as well as of the development of the Otto calculus, and the synthetic Ricci bounds of Lott-Sturm-Villani. The workshop provided participants with an opportunity to discuss how these techniques can be used to approach problems in optimal transport for non-local operators, subelliptic operators in finite and infinite dimensions, analysis on singular spaces, as well as random walks in random media