Goal: To understand and demonstrate the robustness and limitations of the LLN and CLT under heavy-tailed distributions through simulation-based analysis.
This project explores the empirical behaviour of the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT) on both light-tailed and heavy-tailed distributions. Through extensive simulations, I examined how these foundational theorems perform under well-behaved distributions such as Poisson, and more challenging ones like Cauchy and Pareto.
LLN Observations:
- Confirmed convergence for Poisson-distributed sample means
- Demonstrated LLN breakdown with Cauchy samples due to infinite variance
CLT Exploration:
- Simulated sample means from Pareto(5) and compared their distribution to a Normal approximation
- Generated histograms and overlaid theoretical CLT-predicted PDFs
- Used QQ plots to measure convergence and identify heavy-tail effects
- Highlighted failure of the CLT on Pareto(2) due to undefined variance
Tech stack: Python, NumPy, SciPy, matplotlib