# The probability that the minimum of a multivariate Gaussian exceeds zero

Cross Validated Asked by jld on November 26, 2020

Suppose $$X sim mathcal N_n(text{diag}(Sigma), sigma^2 Sigma)$$ where $$Sigma$$ may be allowed to be low rank, and let $$Y = min_i X_i$$. What can be said about $$Pleft(Y geq 0right)$$?

In general I know that the exact distributions of Gaussian order statistics can be intractable, such as this math.se Q&A and the discussion here, but I’m hoping that the relationship between the mean and covariance matrix may lead to some simplification, or how I don’t need the distribution of $$Y$$ but rather just the probability that it is greater than zero. The $$X_i$$ not being iid prevents me from using the usual things I know for examining minima and maxima but I’m still hoping something can be done aside from numerical integration/simulation given values of $$Sigma$$ and $$sigma$$. I’d be very interested in approximations too.

The context on this and the unusual mean vector come from a now-deleted question on stats.se that essentially asked the following: if we have $$Xsimmathcal N_k(mathbf 0, sigma^2 I)$$ and nonrandom nonzero vectors $$z_1,dots,z_ninmathbb R^k$$, what is the probability that $$|X|^2 leq |X-z_i|^2$$ for all $$i$$?

$$|X-z_i|^2 = |X|^2 – 2 X^Tz_i + |z_i|^2$$ so the question is equivalent to $$P(|z_i|^2- 2 X^Tz_i geq 0 text{ for all }i)$$. I collected the $$z_i$$ into the columns of a $$ktimes n$$ matrix $$Z$$ so I can write the random variables in question as an affine transformation of $$X$$ via
$$text{diag}(Z^TZ) – 2 Z^TX sim mathcal N_n(text{diag}(Z^TZ), 4sigma^2 Z^TZ)$$
and I want the probability that this random vector is all non-negative, so this led me to the question I asked. The factored form of $$Sigma$$ here is why I want to allow for possibly low rank covariance matrices since I could have $$k leq n$$.

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