Suppose $X_1$ and $X_2$ are two independent $N(\mu, K)$ Gaussian random vectors in $\mathbb{R}^n$. If we denote by $i_1^*$ the coordinate at which $X_1$ has maximum value, I want to get a lower bound on the following probability $P( X_2(i_1^*)>X_1(i_1^*))$.
I think since $i_1^*$ is also a random variable, my approach is to first condition it on $X_1$ which fixes $i_1^*$. After that we can write
$P(X_2(i_1^*)>X_1(i_1^*)) \geq \mathbb{E}[P(X_2(i_1^*)>X_1(i_1^*)|X_1) ] \geq \mathbb{E}[P(X_2(i_1^*)>c, X_1(i_1^*)\leq c|X_1) ]$ for some $c >0$.
Is there a better way of going about doing this. I think that it should be bounded away from zero, and possibly the probability would be small if $n$ increases. Also, I don't think I have used the fact that the two have the same distribution anywhere. Thanks.