Cramer type moderate deviations for random fields and mutual information estimation for mixed-pair random variables

In this dissertation we first study Cramer type moderate deviation for partial sums of random fields by applying the conjugate method. In 1938 Cramer published his results on large deviations of sums of i.i.d. random variables after which a lot of research has been done on establishing Cramer type moderate and large deviation theorems for different types of random variables and for various statistics. In particular results have been obtained for independent non-identically distributed random variables for the sum of independent random to estimate the mutual information between two random variables. The estimates enjoy a central limit theorem under some regular conditions on the distributions. The theoretical results are demonstrated by simulation study. variables with p-th moment (p \u3e 2) and for different types of dependent random variables. In this work we establish Cramer type exact moderate deviation theorem for random fields. We then show that obtained results are applicable to the partial sums of linear random fields with short or long memory and to non-parametric regression with random field errors. We also show that the result for linear random fields can be applied to calculate the tail probability of partial sums of various models such as the autoregressive fractionally integrated moving average FARIMA(p;\beta; q) processes. The results can also be used to approximate the risk measures such as quantiles and tail conditional expectations of time series or spacial random fields. We also study the mutual information estimation for mixed-pair random variables. One random variable is discrete and the other one is continuous. We develop a kernel metho

Moment inequalities for sums of weakly dependent random fields

We derive both Azuma-Hoeffding and Burkholder-type inequalities for partial sums over a rectangular grid of dimension $d$ of a random field satisfying a weak dependency assumption of projective type: the difference between the expectation of an element of the random field and its conditional expectation given the rest of the field at a distance more than $\delta$ is bounded, in $L^p$ distance, by a known decreasing function of $\delta$. The analysis is based on the combination of a multi-scale approximation of random sums by martingale difference sequences, and of a careful decomposition of the domain. The obtained results extend previously known bounds under comparable hypotheses, and do not use the assumption of commuting filtrations.Comment: 20 pages, 3 figure