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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