Electronic Theses (Proquest)

Permanent URI for this collection

Now showing 1 - 1 of 1

Shannon-McMillan-Breiman Theorem on R
Nguyen, May
Entropy is an important mathematical property in quantifying the amount of uncertainty in a random source. One of the fundamental results in the study of entropy is the Shannon-McMillan-Breiman theorem [2, 13, 16]. In its classical form, for an ergodic stationary process on countable spaces, it asserts that the exponential convergence of the measure of cylinder sets at a rate equal to the entropy. The analog of the Shannon-McMillan-Breiman theorem for stationary process on non-countable spaces was developed over twenty years and completed in the 80’s. However, its analog for random walks on real numbers remained unsolved in the last forty years, until in recent result [6]. This thesis aims to explore possible solutions to complete the entropy theory of random walks on real numbers by proving strong versions of the Shannon-McMillan-Breiman theorem such as almost surely and L¹ convergence. The structure of thesis follows. Chapter 1 is devoted to necessary background in probability theory, ergodic theory, and information theory. Chapter 2 includes the theory of random walks on integers, and the proof of Shannon- McMillan-Breiman theorem on Z. Chapter 3 discusses random walks on real numbers, and we prove strong versions of Shannon- McMillan-Breiman theorem for a vast class of random walks on real numbers.