Tail inequalities for multivariate normal distribution

Question

There exists an closed expression for univariate normal CDF, together with simpler upper-bounds under the form, $$ \Pr\big[X > c\big] \leq \frac{1}{2}\exp\Big(\frac{-c^2}{2}\Big)~, $$ $$\text{where } X \sim \mathcal{N}(0,1)~.$$ Even if there are some algorithms to compute the CDF for multivariate normal distribution, there is no analytical formula for multivariate CDF (as mentioned in this thread). Though, is there some equivalent upper bounds for multivariate normal distribution, $$\Pr\big[X_1>C_1 \land \dots \land X_k>C_k\big] \leq ~?$$ $$\text{where } (X_1, \dots, X_n) \sim \mathcal{N}(\mu, \Sigma)~.$$

Answer 1

These inequalities have been studied by [Savage1962]:

Let $M=\Sigma^{-1}$ with $M = (m_{ij})$. Assuming that for all $1\leq i\leq d$, we have $\Delta_i := \sum_{j=1}^d C_j m_{ji} > 0$, then, $$F(C,\Sigma) = \frac{ |M|^{\frac 1 2}}{(2\pi)^{\frac d 2}} \int_0^\infty\cdots\int_0^\infty \exp\Big[-\frac 1 2 (X+C)^\top M(X+C) \Big] dX_1\dots dX_d$$ $$ < \Big(\prod_{i=1}^{d} \Delta_i\Big)^{-1} \frac{ |M|^{\frac 1 2}}{(2\pi)^{\frac d 2}} \exp \Big[ - \frac 1 2 C^\top M C \Big]~.$$

And more recently by [Hashorva2003] in a more general case.

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[Savage1962] Savage, I.R. Mill's ratio for multivariate normal distributions. J. Res. Natl. Bur. Stand., Sec. B: Math.& Math. Phys., Vol. 66B, No. 3, p. 93 (1962)

[Hashorva2003] Hashorva, E, and Hüsler, J. On multivariate Gaussian tails. Annals of the Institute of Statistical Mathematics 55.3 p. 507-522. (2003)