Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.

What are some nontrivial bounds on $p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$?

This is motivated by the inequality

$$\mathbb{E}(|Z_1|\mathbf{1_S})\ge \max\bigg(0,p\mathbb{E}|Z_1|-\sqrt{\mu_i^2+\Sigma_{ii}-(\mathbb{E}|Z_1|)^2}\sqrt{p-p^2}\bigg)$$

Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.

What are some nontrivial bounds on $p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$?

This is motivated by the inequality

$$\mathbb{E}(|Z_1|\mathbf{1_S})\ge \max\bigg(0,p\,\mathbb{E}|Z_1|-\sqrt{\mu_i^2+\Sigma_{ii}-(\mathbb{E}|Z_1|)^2}\sqrt{p-p^2}\bigg)$$

Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.

What are some nontrivial bounds on $p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max Z$?

This is motivated by the inequality

$$\mathbb{E}(|Z_1|\mathbf{1_S})\ge \max\bigg(0,p\mathbb{E}|Z_1|-\sqrt{\mu_i^2+\Sigma_{ii}-(\mathbb{E}|Z_1|)^2}\sqrt{p-p^2}\bigg)$$

Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.

What are some nontrivial bounds on $p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$?

This is motivated by the inequality

$$\mathbb{E}(|Z_1|\mathbf{1_S})\ge \max\bigg(0,p\mathbb{E}|Z_1|-\sqrt{\mu_i^2+\Sigma_{ii}-(\mathbb{E}|Z_1|)^2}\sqrt{p-p^2}\bigg)$$

Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.

What are some nontrivial bounds on $p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max Z$?

This is motivated by the inequality

$$\mathbb{E}(|Z_1|\mathbf{1_S})\ge \max\bigg(0,p\mathbb{E}|Z_1|-\sqrt{\mu_i^2+\Sigma_{ii}}\sqrt{p-p^2}\bigg)$$

Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.

What are some nontrivial bounds on $p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max Z$?

This is motivated by the inequality

$$\mathbb{E}(|Z_1|\mathbf{1_S})\ge \max\bigg(0,p\mathbb{E}|Z_1|-\sqrt{\mu_i^2+\Sigma_{ii}-(\mathbb{E}|Z_1|)^2}\sqrt{p-p^2}\bigg)$$