A note on the rate of convergence in the strong law of large numbers for martingales

We prove a Baum-Katz-Nagaev type rate of convergence in the Marcinkiewicz-Zygmund and Kolmogorov strong laws of large numbers for norm bounded martingale difference sequences.     

A CONTINUOUS TIME KRONECKER'S LEMMA AND MARTINGALE CONVERGENCE

A continuous time version of Kronecker's lemma is established. This is used to obtain convergence results for martingales.     

On oblique reflecting switched diffusion processes

In this paper, we construct an oblique reflecting switched diffusion process and give the Skorohod representation for this process by using the Dirichlet space theory. We show the tightness property for a sequence of these processes. To do this, we extend the Lyons–Zheng decomposition theorem to the oblique reflecting switched diffusion process     

On z-analogue of Stepanov–Lomonosov–Polesskii inequality

Stepanov?s inequality and its various extensions provide an upper bound for connectedness probability for a Bernoulli-type random subgraph of a given graph. We have found an analogue of this bound for the expected value of the connectedness-event indicator times a positive z raised to the number of edges in the random subgraph. We demonstrate the power of this bound by a quick derivation of a relatively sharp bound for the number of the spanning connected, sparsely edged, subgraphs of a high-degree regular graph.     

Martingale expansion in mixed normal limit

The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived by Yoshida [28] as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a martingale with asymptotically mixed normal distribution. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. Applications to a quadratic form of a diffusion process (“realized volatility”) are discussed.     

Approximations for chance-constrained programming problems

This paper proposes an approximation approach to the solution of chance-constrained stochastic programming problems. The results rely in a fundamental way on the theory of convergence of sequences of measurable multifunctions. Particular results are presented for stochastic linear programming problems.     

A note on two measures of dependence

In a paper in 1956, Dobrushin proved a central limit theorem for triangular arrays of Markov chains under certain dependence assumptions somewhat related to “Doeblin’s condition”. In a very recent paper, Peligrad proved a central limit theorem of the same type, but with dependence assumptions based on maximal correlations. This note here will give a concrete example to illustrate the extent to which Peligrad’s result goes beyond that of Dobrushin.     

On the Cauchy Problem of Evolution p-Laplacian Equation with Nonlinear Gradient Term

The authors study the existence of solution to p-Laplacian equation with non-linear forcing term under optimal assumptions on the initial data,which are assumed to be measures.The existence of local solution is obtained.