Compare LLN and CLT with heavy tails

Q: Compare LLN and CLT with heavy tails

This question evaluates understanding of probabilistic limit theorems and heavy-tailed behavior, specifically the assumptions and convergence guarantees of the Law of Large Numbers and the Central Limit Theorem when applied to Pareto-like distributions.

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Question

Explain the Law of Large Numbers vs the Central Limit Theorem, including their assumptions and convergence guarantees. Construct a concrete counterexample using a Pareto(α=1.5, xm=1) distribution: (a) Does the LLN hold for the sample mean? (b) Does the classical CLT hold for the standardized sample mean? (c) What limiting behavior do you expect for the properly scaled sum? Justify rigorously, and outline a simulation to empirically verify your claims.

Compare LLN and CLT with heavy tails

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