# What is the closed-form or the deterministic form of a quadratic form probability inequality

Hello, everyone, I want to resolve one optimal problem, with the following probability inequality constraint. $Pr(h^H(W_1 - W_2 -W_3 -U)h \geq \sigma^2) \leq \rho$

where $h \sim CN(0,I) \ \text{is random vector}, I \ \text{is a identity matrix};\ w_i \in C^{L\times 1}, W_i = w_iw_i^H; \$ $U$ is a idenpotent matrix,i.e., $U^2 = U\in H^{L \times L} \ \text{and symmetric},\text{Rank}(U)=r < L.$

I find that the probability inequality can be converted into a deterministic form using the Proposition 1.1 in link text. I want to know whether it has a closed-from expression and how to get it, or whether there is another method that can be used to convert it as a deterministic form?

P.S. I know that $h^HW_ih$ following the exponential distribution, and $h^HUh$ following the $\chi^2$ distribution with $r$ degrees of freedom. But they are not independent, because the $w_i$ is the optimal variable, and they are related with each other in the optimal problem.

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