Lyapunov Central Limit Theorem: Theoretical Properties and Applications in Big-Data-Populated Smart City Settings

Central Limit Theorems have a fundamental role in statistics and in a wide range of practical applications. The most famous formulation was proposed by Lindeberg–Lévy and it requires the variables to be independent and identically distributed. In the real setting these conditions are rarely matched, though. The Lyapunov Central Limit Theorem overcomes this limitation, since it does not require the same distribution of the random variables. However, the cost of this generalization is an increased complexity, moderately limiting its effective applicability. In this paper, we resume the main results on the Lyapunov Central Limit Theorem, providing an easy-to-prove condition to put in practice, and demonstrating its uniform convergence. These theoretical results are supported by some relevant applications in the field of big data in smart city settings.