Why this part matters
Exact sampling distributions are often unavailable. Large-sample theory supplies two different guarantees: consistency says an estimator approaches its target, while asymptotic normality describes its fluctuations at a specified scale. Neither claim should be read as a guarantee for every sample size or every loss function.
Laws of large numbers, the CLT, Slutsky and the delta method connect the estimators in Part II to uncertainty calculations. These tools support later confidence intervals and tests in Part VII and STAT 710. The final lecture also explains why tail risk, small samples and growing dimension require care.
Learning objectives
Distinguish convergence modes and use stochastic-order notation (Course outcome 1).
Prove consistency from laws of large numbers and continuity (Course outcome 4).
Apply a CLT with its correct centering and normalization (Course outcome 4).
Use Slutsky and the delta method to derive a limiting distribution (Course outcome 2).
Explain limits of an asymptotic approximation (Course outcome 5).
Roadmap
| Lecture | Purpose |
| 9. Convergence, laws of large numbers, and consistency | Use laws of large numbers to establish consistency. |
| 10. Weak convergence and the central limit theorem | Describe weak convergence and Normal fluctuations with the CLT. |
| 11. Asymptotic normality and the delta method | Transfer limit laws through continuous and differentiable maps. |
What you need from earlier lectures
Where this leads
Optimality returns to finite samples, while high-dimensional theory studies regimes where classical approximations can fail. The convergence tools also support inference in STAT 710. Continue to Part IV.
Readings
Use the lecture notes together with the course texts and original notes. Exact textbook sections and proof references are linked in the lectures.
Part checkpoint
Try each question before opening its answer. This self-check reviews the linked material and adds no exam requirements; the lecture priorities and exceptions still apply.
Does convergence in probability imply almost-sure convergence?
Not in general. Convergence in probability controls each marginal error probability; almost-sure convergence requires eventual control along almost every complete sample path. Return to the explanation.
What is the fluctuation scale of an i.i.d. sample mean with positive finite variance?
The centered mean has scale \(n^{-1/2}\): \(\sqrt n(\bar X_n-\mu)/\sigma\) converges in distribution to a standard Normal. Return to the explanation.
When does Slutsky permit replacing a random denominator?
When it converges in probability to a nonzero constant. Marginal convergence to a nonconstant random variable is not enough to determine a ratio limit. Return to the explanation.
Does asymptotic normality imply convergence of mean squared error?
No. Rare large errors can dominate the second moment while disappearing from the distributional limit. Additional moment control is needed. Return to the explanation.
You should now be able to carry out the learning objectives above and explain where each result needs its assumptions.