Large Deviations Principle

This lecture covers the Large Deviations Principle, discussing the asymptotic behavior of sums of independent random variables. It explains the concept through examples and proofs, emphasizing the exponential decay of tail probabilities. The lecture also introduces the Laplace transform and Chebyshev's inequality to analyze deviations from the mean. Additionally, it explores the Complement of the Large Deviations Principle, highlighting the polynomial decay of tail probabilities. The instructor illustrates these principles with mathematical derivations and applications, providing insights into the regularity of characteristic functions and moments of random variables.