Limit Theorems Summary

Probability · Axiom Academy

Let's review the key theorems about convergence and approximations in probability. Weak LLN: Sample averages converge in probability to the population mean as sample size increases Strong LLN: Sample averages converge almost surely to the population mean (stronger guarantee) Intuition: Averaging more observations gives more reliable estimates of the true mean Example: Flipping a coin many times: the proportion of heads approaches 0.5 Key Idea: Sample means are approximately normal for large samples, regardless of the original distribution Distribution: where n is sample size Rule of Thumb: Generally accurate when n ≥ 30 Power: Allows us to use normal distribution tables for any distribution with finite variance Almost Sure: The strongest; sequence converges with probability 1 In Probability: Probability of being far from limit goes to zero In Distribution: CDFs converge; weakest form Hierarchy: Almost sure ⇒ In probability ⇒ In distribution This standardized form approaches the standard normal distribution as n → ∞ Example Recap: Normal Approximation to Binomial Step 1 - Check Conditions: Verify that np ≥ 5 and n(1-p) ≥ 5 to ensure the approximation is accurate Step 2 - Find Parameters: Calculate mean μ = np and variance σ² = np(1-p) for the approximating normal distribution Step 3 - Apply Continuity Correction: Adjust discrete values: P(X ≤ k) becomes P(X ≤ k + 0.5) in the continuous normal

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