A Novel Model for Investigating GABAergic Synapse Formation

This thesis focuses on expositing the Central Limit Theorem (CLT), its applications, and proving some original extensions of the application of the CLT in renewal theory. We begin by proving the classical CLT for independent and identically distributed (i.i.d) random variables. In the second chapter, we extend the classical result to sequences of independent but not necessarily identically distributed random variables. In doing so, we discuss the Lindeberg and Lyapunov conditions and their implications. Lastly, we cover a few applications of the Lindeberg CLT and prove a few original extensions. In particular, we extend the CLT with random indices and the CLT in renewal theory to allow for non-identically distributed random variables that have uniformly bounded variances and satisfies a Lindeberg-type condition for raw moments instead of central moments.