TL;DR
States that as the number of independent trials increases, the sample mean converges to the expected value. The weak form guarantees convergence in probability; the strong form guarantees almost sure convergence.
Law of Large Numbers
States that as the number of independent trials increases, the sample mean converges to the expected value. The weak form guarantees convergence in probability; the strong form guarantees almost sure convergence.
Why it matters for interviews
Underpins the theoretical basis for backtesting strategies over large samples, Monte Carlo convergence, and the reliability of statistical estimators used in quantitative finance.
Definition and Mathematical Foundation
States that as the number of independent trials increases, the sample mean converges to the expected value. The weak form guarantees convergence in probability; the strong form guarantees almost sure convergence.
The law comes in two forms. The weak law (Khintchine) states convergence in probability: for any \( \epsilon > 0 \), \( P(|\bar{X}_n - \mu| > \epsilon) \to 0 \). The strong law (Kolmogorov) states almost sure convergence: \( P(\lim_{n \to \infty} \bar{X}_n = \mu) = 1 \). Both require finite mean; the strong law additionally requires finite variance or independence.
Application in Quantitative Finance
Underpins the theoretical basis for backtesting strategies over large samples, Monte Carlo convergence, and the reliability of statistical estimators used in quantitative finance.
Related Terms
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