Slutsky's theorem

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Slutsky's theorem is a fundamental result in probability theory attributed to Eugen Slutsky.

[edit] The basic theorem

Let $(X n)$ and $(Y n)$ be sequences of univariate random variables. If $(X n)$ converges in distribution to $X$ and $(Y n)$ converges in probability to a constant $c$ , then $(X n + Y n)$ converges in distribution to $X + c$ , $(X n Y n)$ converges in distribution to $c X$ , and $(X n / Y n)$ converges in distribution to $X / c$ if $c \neq 0$ .

[edit] The generalized theorem

With the conditions as before, the sequence $\left(g(X_n,Y_n)\right)$ converges in distribution to $g (X, c)$ for all continuous $g:\mathbb{R}^2\rightarrow\mathbb{R}$ .