#### Date of Award

1-1-2019

#### Document Type

Dissertation

#### Degree Name

Ph.D. in Mathematics

#### First Advisor

Hailin Sang

#### Second Advisor

Gerard Buskes

#### School

University of Mississippi

#### Relational Format

dissertation/thesis

#### Abstract

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 > 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 method

#### Recommended Citation

Beknazaryan, Aleksandr, "Cramer type moderate deviations for random fields and mutual information estimation for mixed-pair random variables" (2019). *Electronic Theses and Dissertations*. 1737.

https://egrove.olemiss.edu/etd/1737