Bounds for expected maxima of Gaussian processes and their discrete approximations

Konstantin Borovkov; Yuliya Mishura; Alexander Novikov; Mikhail Zhitlukhin

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arXiv · 1508.00099

Bounds for expected maxima of Gaussian processes and their discrete approximations

Konstantin Borovkov·Yuliya Mishura·Alexander Novikov·Mikhail Zhitlukhin

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Abstract

The paper deals with the expected maxima of continuous Gaussian processes $X = (X_t)_{t\ge 0}$ that are Hölder continuous in $L_2$-norm and/or satisfy the opposite inequality for the $L_2$-norms of their increments. Examples of such processes include the fractional Brownian motion and some of its "relatives" (of which several examples are given in the paper). We establish upper and lower bounds for $E \max_{0\le t\le 1}X_t$ and investigate the rate of convergence to that quantity of its discrete approximation $E \max_{0\le i\le n}X_{i/n}$. Some further properties of these two maxima are established in the special case of the fractional Brownian motion.

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Konstantin Borovkov, Yuliya Mishura, Alexander Novikov, Mikhail Zhitlukhin. 2015-08-01. Bounds for expected maxima of Gaussian processes and their discrete approximations. https://arxiv.org/abs/1508.00099

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