A backtest mean estimates expected return, and a Monte Carlo mean estimates a model expectation. The law of large numbers explains consistency. The central limit theorem describes the fluctuation of an average around its target. This lesson states their assumptions, works return and loss examples, and compares a guaranteed Chebyshev bound with an approximate normal error calculation.
Learning objectives. By the end of this note you should be able to:
- state the WLLN (and SLLN) with hypotheses, and interpret convergence in probability;
- state the CLT, standardise a sum, and run a normal approximation for a sample mean of returns;
- explain why aggregated independent risks look normal at scale , and where heavy tails or dependence break the approximation;
- contrast Chebyshev-based vs CLT-based sample-size reasoning.
In this lesson
- Sample means and Chebyshev
- The weak and strong laws
- The central limit theorem
- Where the theorems fail for financial data
- Chebyshev vs CLT for sample size
- Standardising a sum of risks
- Connection to estimation
Before you start
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