# random matrices – Using linearization trick (free chance) to compute limiting eccentric-value density of \$R=XY+Z+A\$ (or equivalently, of \$RR^high\$) retort

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## random matrices – Using linearization trick (free chance) to compute limiting eccentric-value density of \$R=XY+Z+A\$ (or equivalently, of \$RR^high\$)

Disclaimer. I solely began erudition the signify of free chance $$1$$ day in the past, and I’m quiet making an attempt to fullfil the basics, whereas making use of them to my avow particular issues arizing within the spectral evaluation of inescapable concrete random matrices.

Let $$X_{n,m}$$, $$Y_{m,okay}$$, $$Z_{m,okay}$$ breathe sizable (sizable within the sense that $$n to infty$$ such that $$m/n,okay/n in (0,infty)$$, say) impartial random matrices with entries from $$N(0,1)$$ and let $$A_{m,okay}$$ breathe a deterministic matrix. assume the random matrix $$R_{n,okay} := X_{n,m}Y_{m,okay}+Z_{m,okay}+A_{m,okay}$$ (or equivalently, of $$R_{n,okay}R_{n,okay}^high$$).

Question. How to make use of instruments from free chance (e.g the “linearization trick”, and many others.) to compute the limiting eccentric-value distribution of $$R_{n,okay}$$ ?

level to. In explicit, (when workable) I’d affection to carry bounds on the extremities of the uphold of this distribution, as that is my final purpose.

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