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Statement

Notation as in Proposition 1: $r_{j,k}$ are the correlations of $Z(t_1),\ldots,Z(t_n)$, and $P(m,h)$ is the probability that $V_j\le(u-r_{j,m}(u+h))/\sqrt{1-r_{j,m}^2}$ for all $j\le m-1$. $\Phi$ is the standard normal distribution function.

Proposition 2 (Comparison step, p. 4). Let $u\ge0$, and let $h$ be small enough that every upper bound $(u-r_{j,m}(u+h))/\sqrt{1-r_{j,m}^2}$ in the definition of $P(m,h)$ is nonnegative. Suppose there are numbers $c_j=c_j(m,h)>0$ and $d_j=d_j(m,h)>0$ such that

• $c_j/d_j$ is nondecreasing in $1\le j\le m-1$, and
• for each pair $1\le j,k\le m-1$, the number $c_{\min\{j,k\}}d_{\max\{j,k\}}$ is a strict lower bound for $r_{j,k}-r_{j,m}r_{k,m}$.

Then for every $\delta\ge0$,

$$P(m,h)\ \ge\ \int_{-B(\delta)}^{B(\delta)}\frac{e^{-t^2/2}}{\sqrt{2\pi}}\,dt \cdot\prod_{j=1}^{m-1}\Phi\Bigl(\frac{(1-\delta)(u-r_{j,m}(u+h))} {\sqrt{1-r_{j,m}^2-c_jd_j}}\Bigr),$$

where

$$B(\delta)=\delta\sqrt{\frac{d_{m-1}}{c_{m-1}}}\ \min_{1\le j\le m-1} \frac{u-r_{j,m}(u+h)}{d_j}.$$

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 4, pp. 13--16; the proof ends on p. 15. A normal comparison inequality (the paper's Comparison Inequality 2, Section 3) and hypothesis (ii) reduce to the case where the covariances of the $V_j$ equal the model values built from $c_{\min\{j,k\}}d_{\max\{j,k\}}$. Such variables are written explicitly as a weighted partial sum of independent normals plus an independent normal term; the partial sums are a Brownian motion sampled at increasing times, and the maximal inequality $\max_{s\le t}W_s\overset{d}{=}|W_t|$ gives the factor involving $B(\delta)$. Section 7 (pp. 25--28) refines the product term for the application to random multiplicative functions.

Dependencies

A normal comparison inequality (Section 3 of the paper); the reflection principle for Brownian motion, quoted from Grimmett and Stirzaker.

Bears on

None of the problem pages directly. With Proposition 1 it gives the paper's Corollary 2, and its Section 7 refinement gives the full range of Corollary 3.