Let $A_n = \sum_{i=1}^n X_iX_i^T$ and $B = X_1X_1^T$ be random matrices of dimension $m\times m$. Note that the elements of $B$ are dependent on elements of $A$. Note that the vectors $X_i$ are iid with respect to each other, while the random variables in a given vector $X_i$ can be dependent on each other.

Let $v$ be a deterministic vector and let $(A_n v)_i$ to refer to the $i$-th element of the vector $A_n v$.

Now consider the vector $A_n B A_n v$, in particular it's $i$ element.

Assume all the expectations below exist for any $n$. Since the matrix $B$ is essentially irrelevant with respect to the asymptotic behavior I want to know if it is valid to say that as $n \to \infty$, $$ \begin{align} E[|(A_n B A_n v)_i|] \le C E[|(A_n A_n v)_i|] = C E[|(A_n^2 v)_i|], \end{align} $$ for some constant $C$? Can we just ignore the $B$ since it is not dependent on $n$? I mainly interested in whether the above proposition holds for $n$ large enough, but if it also can be shown to hold for every $n$ that would be interesting too, although probably much more difficult (?).

Also, I am happy to assume that the random variables in the random vectors $X_i$ are bounded if that makes things easier.

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    $\begingroup$ Did you omit something? There's no information that would enable us to "note" that $B$ depends on $A$ (as you assert in the second sentence). Indeed, why aren't you writing $B_n$ instead of $B$? How does $B$ depend on the $A_n$? Isn't that critical to formulating an answerable question? How, then, can you justify your later assertion that "$B$ ... is not dependent on $n$"? Does "$|A_n|$" refer to the determinant? One might think so, but that would be inconsistent with much of what you assert about rates of growth later. Maybe it's some kind of matrix norm? $\endgroup$ – whuber Mar 31 at 21:24
  • $\begingroup$ $A_n = \sum_{i=1}^n X_iX_i^T$ whereas $B = X_1 X_1^T$, so you can see that $X_1$ appears in both $A_n$ and $B$, but $B$ is independent of $n$. $\endgroup$ – sonicboom Mar 31 at 21:31
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    $\begingroup$ Thank you for adding that. In order to make progress towards an answer, we still need to know what you are assuming about the $X_i.$ For instance, are they supposed to be iid? Do you want $C$ not to depend on $v$? Not to depend on $i$? Do you want your inequalities to hold for every $n$ or just eventually for large enough $n$? $\endgroup$ – whuber Mar 31 at 21:52
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    $\begingroup$ "IId" is crucial, so please state that in the question. Without that assumption no definite answer is possible, while with it I think there is a clear answer. $\endgroup$ – whuber Apr 1 at 14:23
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    $\begingroup$ If the variables are bounded, the problem is trivial. Otherwise, the existence of any $C$ depends on whether the sixth moments of $X$ exist. $\endgroup$ – whuber Apr 2 at 13:34

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