## Date of Award

1-1-2016

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## College/School/Department

Department of Mathematics and Statistics

## Content Description

1 online resource (vii, 33 pages)

## Dissertation/Thesis Chair

Karin Reinhold

## Committee Members

Charles A. Micchelli, Martin Hildebrand, Joshua Isralowitz

## Keywords

erogodic, erogodic average, smooth ergodic avergage, variation inequality, variation norm, Ergodic theory, Time-series analysis, Random variables, Inequalities (Mathematics), Averaging method (Differential equations), Convergence

## Subject Categories

Physical Sciences and Mathematics

## Abstract

The Ergodic Theorem talks about the convergence of time averages of systems. The pointwise Ergodic Theorem states that the time averages converge to the space averages almost everywhere, for any integrable function $f$. When we consider in continuous time, because of instrumental limitations, time measurements cannot be taken exactly at any instant of time. Therefore, instead of dealing with averages along arithmetic sequences, in applications one has a smooth average around the time of observation. In this dissertation we investigate the pointwise behavior of smoothed out average with a measure preserving continuous flow on a probability space, $\displaystyle K_{n} f(x) = \frac{1}{n}\sum_{k=0}^{n-1} \int \varphi_{\varepsilon_{k}} (t) f\left( T_{k+t} x \right) \, dt$ where $\varepsilon_{k}$ are i.i.d positive random variables. We prove a variation inequality for this weighted smoothed average and its convergence a.e. in $L^{2}$ for any realization of the random variable $\varepsilon_{k}$ in a set of probability $1$.

## Recommended Citation

Choi, Jaeyong, "Convergence of a smooth random average and its variation inequality" (2016). *Legacy Theses & Dissertations (2009 - 2024)*. 1584.

https://scholarsarchive.library.albany.edu/legacy-etd/1584