A backtest's average return over 1,000 trading days is treated as a reliable estimate of the strategy's true expected return — but why is a sample average a good estimate of a population mean? The weak law of large numbers answers this precisely: as the sample size grows, the sample mean's probability of deviating from the true mean by any fixed amount shrinks to zero. This note derives the weak law as a direct corollary of Chebyshev's inequality, then formalises exactly what kind of convergence this is — convergence in probability — a genuinely new notion of limit for a sequence of random variables, illustrated with examples that show both what it guarantees and, sharply, what it does not.
Learning objectives. By the end of this note you should be able to:
- derive the weak law of large numbers from the Chebyshev inequality;
- state the formal definition of convergence in probability and contrast it with ordinary (deterministic) convergence;
- verify convergence in probability for a concrete sequence of random variables;
- recognise that convergence in probability does not imply convergence of expectations.
In this lesson
- The weak law of large numbers
- Convergence in probability
Before you start
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