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Non-classical convergence results for sums of dependent random variables

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Degree
Doctor of Philosophy (Ph.D.), Bowling Green State University, Mathematics/Probability and Statistics, .
Abstract

Let X be a random variable with piecewise continuous and bounded probability density function. We consider the sequence of fractional parts of multiples of X. We prove that this sequence of random variables is a strongly mixing sequence and hence is asymptotically independent of any other random variable. This is the basis for the main investigation, which we describe now.

Let T follow the Uniform distribution on (0,1). We consider the sequences of sine and cosine of multiples of T. We show that these random variables are identically distributed, uncorrelated but dependent, non-Markovian and non-exchangeable. To understand the dependence, we investigate the sum of the first n terms of each sequence and let n go to infinity. We would like to derive the asymptotic distribution of sums of these sequences.

In classical convergence results, one either considers ergodic type results, involving dividing the sum by n or looking at rarer subsequences and derive a Normal limit. In our non-classical results, we show in case of the sine sequence, that the sum converges in distribution to the Cauchy distribution without normalization and in the case of the cosine sequence, the limiting distribution of the sum is heavy-tailed and non-normal but not Cauchy. The strong dependence in the sequence is the explanation for why no normalization is needed.

We discuss pointwise convergence and the Cesaro convergence of the corresponding non-random series. We derive the asymptotic distribution of sums of arithmetic subsequences of the sine and cosine sequences. We derive convergence results for sums of weighted sine and cosine sequences and finally we extend our result to derive the asymptotic distribution of sums of a finite linear combination of sine and cosine terms. We discuss some open problems and future research directions.

Mathematics
Keywords
Sums of dependent random variables; Distribution of fractional parts; Non-classical convergence results; Sums of dependent random variables; Probability theory; Trigonometric series; Finite Fourier series