CS31: Extremes, Sojourns and Related Functionals of Gaussian Processes

Organizer: Zbigniew Michna (Wrocław University of Science and Technology)

On a Weak Convergence Theorem for the Normalized Maximum of Stationary Gaussian Processes with a Trend

Goran Popivoda

We present a Gnedeko-type limit theorem, which states that the normalized maximum of the process \(X(t) = \xi(t) - g(t)\), \(t\geq0\), converges weakly to a Gumbel distribution. In this context, \(\xi(t)\) represents a stationary Gaussian process, while \(g(t)\) is a deterministic function. The inclusion of the trend function \(g(t)\) disrupts stationarity, making it challenging to apply classical results.

We provide the normalizing constants \(a_T\) and \(b_T\) such that \(a_T(\max_{t \in [0,T]} X(t) - b_T)\) converges to a mixed Gumbel distribution as \(T \to \infty\). Notably, the normalizing constant \(a_T\) appears to be unaffected by the introduction of the trend, whereas the constant \(b_T\) is influenced by it.

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