Expectation of random matrix (minimum positive eigenvalue)

Assume, $M_i$ are random symmetric positive semi-definite matrices and there exists $ 0 < mu_1 leq mu_2 $ such that $mu_1 |x|^2 leq x^TE[M_i]x leq mu_2 |x|^2$ holds for some $0 neq x in R^n$. Also take $E[M_i] = S^TS$, then we also have $x in textbf{Range}(S^T)$. Now, I am looking for following bounds (any one of them will work): $$ quad E left[frac{|x|^4}{(x^TM_ix)(x^TM_i^2x)}right] leq… quad text{or} quad E left[frac{(x^TM_ix)(x^TM_i^2x)}{(x^Tx)(x^TM_i^3x)}right] geq…$$ We assumed $x^TM_ix, x^TM_i^2x , x^TM_i^3x > 0$. And $E[M_i] = sumlimits_{i=1}^{q} p_i M_i, 0 leq p_i leq 1$.

I tried this for the second, take $l = max_i lambda_{max}(M_i)$, then the second one can be simplified as $geq frac{1}{l} Eleft[frac{x^TM_ix}{|x|^2}right] = frac{1}{l} frac{x^TE[M_i]x}{|x|^2} geq frac{mu_1}{l}$. I was wondering if a tighter bound is possible.

MathOverflow Asked by Ripon on January 20, 2021

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