3.2 Limiting Distributions

3.2.1 Converge in probability

3.2.2 Converge in distribution

Theorem 3.1 Continuous Mapping Theorem

Let $X_n$ be a sequence of random variables such that $X_n \xrightarrow{d} X$, where $X$ is some random variable (or constant). Let $g(\cdot)$ be a continuous function. Then

$$g(X_n) \xrightarrow{d} g(X).$$

If $X$ is a constant $c$, then $g(X_n) \xrightarrow{p} g(c)$. (Convergence in distribution to a constant implies convergence in probability to that constant).

Slutsky’s Theorem is a collection of results concerning the asymptotic behavior of random variables. It is extremely useful in establishing the limiting distributions of estimators and test statistics.

Slutzky theorem

If $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{p} c$, where $c$ is a constant, then If $X_n \xrightarrow{p} X$ and $Y_n \xrightarrow{p} Y$, then

Slutsky’s theorem essentially says that convergence in distribution and convergence in probability behave “nicely” with continuous functions and arithmetic operations, especially when one of the sequences converges to a constant. You can often treat limits of random variables much like limits of ordinary sequences, provided you are careful about the mode of convergence.