Harper 2013 bounds suprema gaussian processes omega results

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corollary_1: Harper shows that there is an absolute constant c > 0, which could be found explicitly, with Pickands constant H_alpha >= c sqrt(alpha)(e alpha/2)^{1/alpha} for all 0 < alpha <= 2, improving the known lower bounds as alpha tends to 0.

corollary_2: For independent standard normal g_p, the probability that the supremum over 1 <= t <= 2(log log x)^2 of the sum over primes p <= x of g_p cos(t log p)/p^{1/2+1/log x} is at most log log x - log log log x + O((log log log x)^{3/4}) is O((log log log x)^{-1/2}) as x tends to infinity.

corollary_3: For the summatory function M(x) of a Rademacher random multiplicative function and any A > 2.5, Harper proves that almost surely M(x) is not O(sqrt(x)(log log x)^{-A}), improving Halász's 1982 omega result M(x) != O(sqrt(x) exp(-B sqrt(log log x log log log x))).

proposition_1: Harper's conditioning step: for centered, unit-variance jointly normal Z(t_1), ..., Z(t_n) with every off-diagonal correlation of absolute value below 1, P(max Z(t_i) > u) is at least H e^{-(u+H)^2/2}/sqrt(2 pi) times the sum over m of the infimum over 0 <= h <= H of an explicit conditioned orthant probability P(m,h), for every u >= 0 and H >= 0.

proposition_2: Harper's comparison step: if the thresholds in P(m,h) are nonnegative and positive c_j, d_j have c_j/d_j nondecreasing with c_{min} d_{max} a strict lower bound for every residual covariance, then for every delta >= 0, P(m,h) is at least P(|N(0,1)| <= B(delta)) times a product of normal distribution values with the variances reduced by c_j d_j.

theorem_1: For a stationary normal sequence Z(t_1), ..., Z(t_n) with decreasing nonnegative correlation r, u >= 1 and r(1)(1 + 2u^{-2}) <= 1, Harper bounds P(max Z(t_i) > u) below by n e^{-u^2/2}/(40u) min{1, sqrt((1 - r(1))/(u^2 r(1)))} times a product of normal distribution values, with an absolute implied constant.

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Harper, Adam J., 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. The copy read for this card is arXiv:1012.0210v2 (22 Feb 2013), an electronic reprint that carries the journal's citation header, prints "© Institute of Mathematical Statistics, 2013" and differs from the original in pagination and typographic detail. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1012.0210), every other right reserved.

Harper develops a non-asymptotic method for lower bounding P(sup_t Z(t) > u) for a Gaussian process: Proposition 1 is a conditioning step reducing the tail to probabilities P(m,h) about conditioned normal vectors, and Proposition 2 is a comparison step that lower bounds P(m,h) by explicitly building a Gaussian family with a prescribed lower-bound correlation structure and applying a Brownian maximal inequality. Theorem 1 combines these for stationary sequences, giving a fully explicit bound valid for moderate u rather than only as u tends to infinity. Corollary 1 applies the machinery to extreme value theory, showing the Pickands constants satisfy H_alpha >= c sqrt(alpha) (e alpha/2)^{1/alpha} for 0 < alpha <= 2. The main application is to the Gaussian analog of Halász's process sum_{p <= x} g_p cos(t log p)/p^{1/2 + 1/log x}: Corollary 2 shows its supremum over 1 <= t <= 2(log log x)^2 exceeds log log x - log log log x + O((log log log x)^{3/4}) except with probability O((log log log x)^{-1/2}), which Harper calls very precise since standard methods bound the supremum by log log x + log log log x with probability 1 - o(1). Transferring this via a multivariate central limit theorem (Appendix B) and Halász's argument (Supplementary Lemma 1, Appendix A) yields Corollary 3 for A > 3, and a sharpening of Proposition 2 by contradiction (Section 7) gives it for all A > 2.5: for a Rademacher random multiplicative function f and M(x) = sum_{m <= x} f(m), almost surely M(x) is not O(sqrt(x)(log log x)^{-A}), improving Halász's 1982 omega result M(x) != O(sqrt(x) exp(-B sqrt(log log x log log log x))) for some B > 0, which the paper calls the best known lower bound result for |M(x)|. Harper writes that M(x) != O(sqrt(x)) almost surely seems extremely likely, perhaps with fluctuations of order sqrt(x log log x) by analogy with the law of the iterated logarithm, or even larger ones since the distribution of M(x) may have heavy tails, and that an argument like Harper's, based on a certain average of M(x), seems unable to detect such large but rare fluctuations.

Source: https://arxiv.org/abs/1012.0210.

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages of arXiv v2; no proof is checked step by step.

Bears on.

• #520: the paper's f is the problem's Rademacher multiplicative function and its M(x) the problem's partial sum. Corollary 3 shows that, for each A > 2.5, almost surely M(x) is not O(sqrt(x)(log log x)^{-A}), a lower bound for the fluctuations far below the scale sqrt(N log log N) in the question, which it does not answer. Page 8 raises fluctuations of order sqrt(x log log x) only as a possibility, by analogy with the law of the iterated logarithm.

Results.

• Proposition 1 (p. 3): Conditioning step: for jointly normal centered unit-variance Z(t_i) with |r_{i,j}| < 1 off the diagonal, and any u, H >= 0, P(max_i Z(t_i) > u) >= (H e^{-(u+H)^2/2}/sqrt(2 pi)) sum_{m=1}^n inf_{0 <= h <= H} P(m,h), with P(m,h) an explicit conditioned normal orthant probability.
• Proposition 2 (p. 4): Comparison step: under nonnegative thresholds and positive c_j, d_j with c_j/d_j nondecreasing and c_{min{j,k}} d_{max{j,k}} a strict lower bound for r_{j,k} - r_{j,m} r_{k,m}, for any delta >= 0, P(m,h) is at least P(|N(0,1)| <= B(delta)) times prod_{j<m} Phi((1-delta)(u - r_{j,m}(u+h))/ sqrt(1 - r_{j,m}^2 - c_j d_j)).
• Theorem 1 (p. 4): For a stationary sequence with decreasing nonnegative correlation r, u >= 1 and r(1)(1+2u^{-2}) <= 1, P(max_i Z(t_i) > u) >= n (e^{-u^2/2}/(40u)) min{1, sqrt((1-r(1))/(u^2 r(1)))} times prod_{j=1}^{n-1} Phi(u sqrt(1-r(j))(1+O(1/(u^2(1-r(j)))))), with an absolute implied constant.
• Corollary 1 (p. 6): There is an absolute constant c > 0, which could be found explicitly, with Pickands constant H_alpha >= c sqrt(alpha) (e alpha/2)^{1/alpha} for all 0 < alpha <= 2.
• Corollary 2 (p. 7): For the Gaussian Halász process sum_{p<=x} g_p cos(t log p) p^{-1/2-1/log x}, the probability that its supremum over 1 <= t <= 2(log log x)^2 is at most log log x - log log log x + O((log log log x)^{3/4}) is O((log log log x)^{-1/2}).
• Corollary 3 (p. 7): For any A > 2.5, the summatory function M(x) of a Rademacher random multiplicative function almost surely satisfies M(x) != O(sqrt(x)(log log x)^{-A}).

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