Convergence of Random Variables: Definitions and Connections

This lecture discusses the convergence of sequences of random variables, introducing three key notions: quadratic convergence, convergence in probability, and almost sure convergence. The instructor begins by defining convergence for sequences of real numbers, emphasizing the simplicity of this concept compared to random variables. The lecture then transitions to the complexities of random variables, explaining that they can converge in various ways. The first notion, quadratic convergence, is defined in the context of L2 space, where the sequence converges if the expectation of the squared difference approaches zero. Next, convergence in probability is introduced, where the probability of deviation from the limit decreases to zero as the sequence progresses. Finally, almost sure convergence is explained, highlighting that it guarantees convergence for almost all outcomes. The instructor also explores the relationships between these notions, illustrating how quadratic convergence implies convergence in probability, while providing examples and conditions under which these relationships hold. The lecture concludes with a preview of further discussions on the connections between these convergence types.