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Let $X_1, X_2$ be two iid random row vectors in $\mathbb{R}^p$, each of whose components are real valued. I'd like to prove that the $\mathbb{R}^{p\times p}$ random matrices $X_1X_1', X_2X_2'$ are also iid, where ' means transpose of a vactor/matrix.

It's clear that these two random matrices are identically distributed, for the $(i,j)$-th entries for them are $X_{1i}X_{1j}, X_{2i}X_{2j}$ respectively, and $X_{1i}, X_{2i}$ are identically distributed, and so are $X_{1j}, X_{2j}$.

But how to prove the independence of these two random matrices?

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That the random matrices $X_1X_1'$ and $X_2X_2'$ are iid is a special case of the following general facts:

Fact 1: Suppose that $X_1$ and $X_2$ are independent random variables (r.v.'s) defined on a probability space $(\Omega,\mathcal F,\mathbb P)$ with values in measurable spaces $(S_1,\mathcal S_1)$ and $(S_2,\mathcal S_2)$. For each $i\in\{1,2\}$, let $g_i$ be a measurable map from $(S_i,\mathcal S_i)$ to a measurable space $(T_i,\mathcal T_i)$. Then the r.v.'s $g_1\circ X_1$ and $g_2\circ X_2$, with values in the measurable spaces $(T_1,\mathcal T_1)$ and $(T_2,\mathcal T_2)$, are independent.

Fact 2: Suppose that $X_1$ and $X_2$ are identically distributed r.v.'s defined on a probability space $(\Omega,\mathcal F,\mathbb P)$ with values in a measurable space $(S,\mathcal S)$. Let $g$ be a measurable map from $(S,\mathcal S)$ to a measurable space $(T,\mathcal T)$. Then the r.v.'s $g\circ X_1$ and $g\circ X_2$, with values in the measurable space $(T,\mathcal T)$, are identically distributed.

Proof of Fact 1: Take any $C_1\in\mathcal T_1$ and $C_2\in\mathcal T_2$. Then $$\mathbb P(g_1\circ X_1\in C_1,g_2\circ X_2\in C_2) :=\mathbb P((g_1\circ X_1)^{-1}(C_1)\cap(g_2\circ X_2)^{-1}(C_1)) =\mathbb P(X_1^{-1}(g_1^{-1}(C_1))\cap X_2^{-1}(g_2^{-1}(C_2))) =\mathbb P(X_1\in g_1^{-1}(C_1),X_2\in g_2^{-1}(C_2)) =\mathbb P(X_1\in g_1^{-1}(C_1))\mathbb P(X_2\in g_2^{-1}(C_2)) =\mathbb P(g_1\circ X_1\in C_1) \mathbb P(g_2\circ X_2\in C_2). $$

Proof of Fact 2: Take any $C\in\mathcal T$. Then $$\mathbb P(g\circ X_1\in C) :=\mathbb P((g\circ X_1)^{-1}(C)) =\mathbb P(X_1^{-1}(g^{-1}(C))) =\mathbb P(X_1\in g^{-1}(C)) =\mathbb P(X_2\in g^{-1}(C)) =\mathbb P(X_2^{-1}(g^{-1}(C))) =\mathbb P((g\circ X_2)^{-1}(C)) =\mathbb P(g\circ X_2\in C). $$

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  • $\begingroup$ Thank you for your answer, much appreciated. Could you recommend a textbok from where we could pikc up stuff like this? I know measure theory and part of measure-theoretic probability. $\endgroup$
    – Learning math
    Commented Oct 25, 2019 at 18:22
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    $\begingroup$ @Let'stalkmath : I think this can be found in some of the standard textbooks on probability theory. At this point, I don't have specific references to Facts 1 and 2 exactly. For me, such facts are much easier to prove than to find in the literature. Fact 1 can be proved a bit simpler, at least in the case when the measurable spaces are $\mathbb R^d$'s (as in your case), using a characterization of independence in terms of the expectation such as e.g. Lemma 6.12 in the text Probability by Davar Khoshnevisan. $\endgroup$
    – Iosif Pinelis
    Commented Oct 25, 2019 at 18:43
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    $\begingroup$ Fact 1 corresponds to condition c) in Theorem 10.1 of Probability Essentials by Jacod and Protter, second ed., ISBN 3-540-43871-8. $\endgroup$
    – Iosif Pinelis
    Commented Oct 25, 2019 at 20:14
  • $\begingroup$ thanks for pointing the reference out! $\endgroup$
    – Learning math
    Commented Oct 25, 2019 at 20:31

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