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I got the following expression, representing the r.h.s. of a tail bound on $Pr[\|Q^Tp\|_2^2 > tk]$, where the $k\times r$ matrix $Q$ has i.i.d. standard Gaussian entries, the vector $p$ is drawn with i.i.d. standard Gaussian entries independently from $Q$, and $t\ge C\cdot r$ for some constant $C>0$:

$(\frac{e\sqrt{tk\rho}}{r+k})^{\frac{r+k}{2}}\cdot\exp(-\frac{\sqrt{tk}}{2})$

where $t,k,\rho,r>0, k< \rho$, and $r$ can be unbounded.

Is it possible to upper bound this with an expression of the form:

$C \cdot t^{-(1+\epsilon)}$

for some constants $C,\epsilon>0$, for all $t\ge C\cdot r$, independently of $r$? (possibly under some conditions?)

Or, can a bound of this for be derived in some other way?

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