Extremes (2015) 18:99–108 DOI 10.1007/s10687-014-0205-x On probability of high extremes for product of two independent Gaussian stationary processes Vladimir I. Piterbarg · Alexander Zhdanov Received: 16 June 2014 / Revised: 13 August 2014 / Accepted: 19 August 2014 / Published online: 31 August 2014 © Springer Science+Business Media New York 2014 Abstract Let X(t), Y(t), t ≥ 0, be two independent zero-mean stationary Gaus- sian processes, whose covariance functions are such that r i (t) = 1 −|t | a i + o(|t | a i ) as t → 0, with 0 <a i ≤ 2, i = 1, 2 and both of the functions are less than one for non-zero t . We derive for any p the exact asymptotic behavior of the probabil- ity P(max t ∈[0,p] X(t)Y(t) > u) as u →∞. We discuss possibilities generalizing obtained results to other Gaussian chaos processes h(X(t)), with a Gaussian vector process X(t) and a homogeneous function h of positive order. Keywords Gaussian processes · Gaussian chaos · High extremes probabilities · Double sum method AMS 2000 Subject Classifications Primary 60G15 · Secondary 60K30 · 60K40 · 60G70 1 Introduction We study in this contribution probabilities of high extremes for product of two inde- pendent Gaussian processes. In Hashorva et al. (2013), Hashorva et al. (2014), tail distributions for Gaussian chaos have been studied, that is probabilies P (h(X) >u) for large u, where X is a Gaussian vector and h is a homogeneous function of positive order. Next steps of our program is studying Gaussian chaos processes, V. I. Piterbarg () · A. Zhdanov Moscow Lomonosov State University, Moscow, Russia e-mail: piter@mech.math.msu.su