Extreme-Value Analysis of Standardized Gaussian Increments

Let {Xi, i = 1, 2, . . .} be i.i.d. standard gaussian variables. Let Sn = X1 + . . . + Xn be the sequence of partial sums and Ln = max 0≤i<j≤n Sj − Si √ j − i . We show that the distribution of Ln, appropriately normalized, converges as n → ∞ to the Gumbel distribution. In some sense, the the random variable Ln, being the maximum of n(n+1)/2 dependent standard gaussian variables, behaves like the maximum of Hn log n independent standard gaussian variables. Here, H ∈ (0,∞) is some constant. We also prove a version of the above result for the Brownian motion.