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Statement

Proposition 1 (Conditioning step, p. 3). Let $Z(t_1),\ldots,Z(t_n)$ be jointly multivariate normal, write $r_{i,j}=\mathbb E\,Z(t_i)Z(t_j)$, and assume that $\mathbb E Z(t_i)=0$ and $\mathbb E Z(t_i)^2=1$ for all $1\le i\le n$, and that $|r_{i,j}|<1$ whenever $i\ne j$. Then for every $u\ge0$ and every $H\ge0$,

$$\mathbb P\Bigl(\max_{1\le i\le n}Z(t_i)>u\Bigr)\ \ge\ \frac{He^{-(u+H)^2/2}}{\sqrt{2\pi}}\sum_{m=1}^{n}\ \inf_{0\le h\le H}P(m,h),$$

where

$$P(m,h)=\mathbb P\Bigl(V_j\le\frac{u-r_{j,m}(u+h)}{\sqrt{1-r_{j,m}^2}} \ \ \forall j\le m-1\Bigr),$$

and the $V_j=V_{j,m}$ are centered, unit-variance, jointly normal random variables with correlations

$$\frac{r_{j,k}-r_{j,m}r_{k,m}}{\sqrt{(1-r_{j,m}^2)(1-r_{k,m}^2)}}.$$

Source. Adam J. Harper, Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function, Ann. Appl. Probab. 23 (2013), no. 2, 584--616, DOI 10.1214/12-AAP847. Labels and pages here are those of the electronic reprint arXiv:1012.0210v2 (22 Feb 2013), whose pagination differs from the journal's. The edition read is identified on the source card.

Read depth. Claims checked: the statement and its hypotheses were read clause by clause on the printed page. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 2, pp. 8--9. The event that the maximum exceeds $u$ splits according to the first index $m$ with $Z(t_m)>u$. The variable $Z(t_m)$ is independent of the residuals $Z(t_j)-r_{j,m}Z(t_m)$, $j\le m-1$, whose correlations are $r_{j,k}-r_{j,m}r_{k,m}$; conditioning on $Z(t_m)=x$ for $u<x\le u+H$ and bounding the normal density below by its value at $u+H$ gives each term. The paper mentions (p. 3) an earlier, more involved proof through a reversal of roles in the normal comparison procedure, described in Section 3.

Dependencies

None beyond elementary properties of the multivariate normal distribution.

Bears on

None of the problem pages directly. It is one of the two ingredients of the paper's lower bound in Corollary 2, which leads to Corollary 3.