Convergence of maximum of sequences of random variables

by Baham91   Last Updated August 14, 2019 00:20 AM

Let $$X_1^n,\ldots,X_k^n$$ be sequences of random variables, where $$k$$ is fixed. The sequences are not independent. I'd like to prove the following statement:

Assume that $$X_1^n \to x_1, \ldots, X_k^n \to x_n$$ in probability, where $$x_1,\ldots,x_n$$ are constants. Then: $$\max_{i \in \{1,\ldots,k\}} X_i^n \to \max_{i \in \{1,\ldots,k\}} x_i$$ in probability.

Notice that this setting is different than the usual one, where the maximum is usually taken over $$n$$.

I'm not entirely sure if this can be seen as a direct application of the continuous mapping theorem, especially if the sequences live in other spaces than $$\mathbb{R}$$ (for instance on $$\mathbb{N}$$).

Any insight is very much appreciated!

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It is indeed just the ($$k$$-dimensional) continuous mapping theorem, since $$F(x_1, \dots, x_k) := \max(x_1, \dots, x_k)$$ is a continuous function from $$\mathbb{R}^k$$ to $$\mathbb{R}$$ (easy exercise). You have a sequence of $$\mathbb{R}^k$$-valued random vectors $$\mathbf{X}^n := (X_1^n, \dots, X_k^n)$$ converging in probability to a constant vector $$\mathbf{x} = (x_1, \dots, x_k)$$, and you are claiming that $$F(\mathbf{X}^n) \to F(\mathbf{x})$$ in probability. That is exactly the continuous mapping theorem.

It makes no difference if your random variables only take values in a subset of $$\mathbb{R}$$, such as $$\mathbb{N}$$ or $$\mathbb{Z}$$ or what have you. You can still view them as real-valued random variables.

(Of course, if your random variables take values in some space completely different from $$\mathbb{R}$$, then you need to know quite a bit about the order and topological structure of that space to even make sense of the statement.)

Nate Eldredge
August 13, 2019 23:10 PM

It is easy to verify that $$|max \{a_1,a_2,...,a_k) - max \{b_1,b_2,...,b_k)|>\epsilon$$ implies $$|a_i-b_i| >\epsilon$$ for some $$i$$. [Use proof by contradiction]. Hence the result follows immediately from definition of convergence in probability.

Kavi Rama Murthy
August 13, 2019 23:28 PM