convergence of random variables
1. types of convergence
2. law of large numbers
The difference between the strong and weak law of large numbers comes from the difference between convergence almost surely and convergence in probability.
3. continuous functions
If \(f\) is a continuous function then, if \(Y_n\) converges to \(Y\) (a.s. OR in probability OR in distribution) then \(f(Y_n)\) converges to \(f(Y)\)
4. marginal convergence
If \(X_n\) converges (a.s. OR in probability) and \(Y_n\) converges (a.s. OR in probability) then \((X_n, Y_n)\) converges to (X, Y)
BUT: \(X_n \overset{d}{\rightarrow} X\) and \(Y_n \overset{d}{\rightarrow} Y\) do not imply \((X_n, Y_n) \overset{d}{\rightarrow} (X, Y)\)
4.0.1. example
Let \(X_1 = X_2 = \cdots = X_n\) where \(X_1 \sim \mathcal{N}(0,1)\) and \(X\) is an independent Gaussian \(\mathcal{N}(0,1)\). We see that the \(X_n\) 's converge in distribution to \(X\), because they both have the same distribution
Let \(Y_1 = -X_1 = Y_2 = \cdots = Y_n\) where \(Y_1 \sim \mathcal{N}(0,1)\) and \(Y\) is an independent Gaussian \(\mathcal{N}(0,1)\). Similarly, the \(Y_n\) 's converge to \(Y\).
But for \((X_n, Y_n)\), \(X_n + Y_n = 0\) but \(X+Y \sim \mathcal{N}(0,2)\).
5. Slutsky's theorem
If \(X_n \overset{d}{\rightarrow} X\) and \(Y_n \overset{d}{\rightarrow} c\) for some constant \(c\). Then, \((X_n, Y_n) \overset{d}{\rightarrow} (X, c)\)
5.1. consequences
- \(X_n + Y_n\) converges in distribution to \(X+c\)
- \(X_nY_n\) converges in distribution to \(Xc\)