CS31: Extremes, Sojourns and Related Functionals of Gaussian Processes

date: 7/17/2025, time: 16:00-17:30, room: ICS 119

Organizer: Zbigniew Michna (Wrocław University of Science and Technology, Department of Operations Research and Business Intelligence)

Chair: Zbigniew Michna (Wrocław University of Science and Technology, Department of Operations Research and Business Intelligence)

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

Goran Popivoda (University of Montenegro, Faculty of Science and Mathematics)

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.

Bibliography

\([1]\) V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields. Providence, Rhode Island: AMS Translations of Mathematical Monographs 148, 1 ed., 1996.

\([2]\) J.-M. Azais and M. Wschebor, Level Sets and Extrema of Random Processes and Fields. New York: Wiley, 1 ed., 2009.

\([3]\) M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics, Springer Verlag, 1983.

\([4]\) V. Piterbarg and S. Stamatovic, “Limit Theorem for High Level a-Upcrossings by \(\chi\)-Process,” Theory of Probability & Its Applications, vol. 48, no. 4, pp. 734–741, 2004.

\([5]\) B. Stamatovic and S. Stamatovic, “Cox limit theorem for large excursions of a norm of a Gaussian vector process,” Statistics & Probability Letters, vol. 80, no. 19, pp. 1479–1485, 2010.

\([6]\) J. Xiao, Y. Wen, and Z. Tan, “The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models,” Statistics & Probability Letters, vol. 149, pp. 38–46, 2019.

\([7]\) D. G. Konstantinides, V. Piterbarg, and S. Stamatovic, “Gnedenko-Type Limit Theorems for Cyclostationary \(\chi^2\)-Processes,” Lithuanian Mathematical Journal, vol. 44, no. 2, pp. 157–167, 2004.

\([8]\) Z. Tan and E. Hashorva, “Limit theorems for extremes of strongly dependent cyclo-stationary \(\chi\)-processes,” Extremes, vol. 16, no. 2, pp. 241–254, 2013.

\([9]\) J.-M. Azais and C. Mercadier, “Asymptotic Poisson Character of Extremes in Non-Stationary Gaussian Models,” Extremes, vol. 6, no. 4, pp. 301–318, 2003.

\([10]\) L. Bai, K. Dȩbicki, E. Hashorva, and L. Ji, “Extremes of threshold-dependent Gaussian processes,” Science China Mathematics, vol. 61, no. 11, pp. 1971–2002, 2018.

\([11]\) E. Hashorva and L. Ji, “Piterbarg theorems for chi-processes with trend,” Extremes, vol. 18, pp. 37–64, 2015.

Extreme Sojourns of Vector-valued Gaussian Processes with Trend

Long Bai (Xi’an Jiaotong-Liverpool University)

Abstract: Let \({\bf X}(t)=(X_1(t), \cdots, X_n(t)),t\in \mathbb{R}\) be a centered vector-valued Gaussian process with independent components and continuous sample paths. For a compact subset \(E\subset \mathbb{R}\), we investigate the asymptotics of \[\begin{aligned} \mathbb{P}\left\{v(u)\int_E\mathbb{I}\left\{\min_{1\le i\le n}({X}_i(t)-c_it)>u\right\}dt>x\right\},\quad x\ge0,\end{aligned}\] as \(u\to\infty\), where \(\mathbb{I}\{\cdot\}\) is the indicator function, \(v(\cdot)\) is a positive scaling function and \(c_i\in \mathbb{R}\) are some trend constants. Specifically, we analyze two fundamental classes of \({\bf X}(t)\): (1) processes with locally stationary components, and (2) processes with non-stationary components. In addition, we derive the asymptotic distributional properties of \[\begin{aligned} \tau_{u}(x):=\inf\left\{t:v(u)\int_0^t\mathbb{I}\left\{\min_{1\le i\le n}({X}_i(s)-c_is)>u\right\}ds>x\right\},\quad x\ge0,\end{aligned}\] as \(u\to\infty\), where \(\inf\emptyset=\infty\).

Non-simultaneous ruin for positively correlated multi-dimensional Brownian motion

Konrad Krystecki (University of Wrocław)

For two-dimensional set parameter \(\mathscr{T}=[0,1]^2\) the non-simultaneous ruin probability can be defined as \[\mathbb{P}(\exists_{s,t \in [0,1]} W_1(s)-c_1s>u,W_2(t)-c_2t>au)\] with \(W_1,W_2\) correlated standard Brownian motions. Exact results for this model were given in \([1]\), but as \([2]\) points out, they are computationally ineffective and are not translatable to higher dimensions. Additionally, in \([3]\) bounds can be found for two-dimensional model with no drifts. Asymptotic results for the two-dimensional non-simultaneous model were given in \([4]\) for infinite time interval and in \([5]\) for finite time interval. This contribution aims at extending the known results to higher dimension by introducing ruin probability \[\mathbb{P}( \exists \vec{t} \in [0,1]^d: \vec{W}(\vec{t})-\vec{ c} \cdot \vec{ t}> \vec{ \alpha} u),\] where \(\vec{ W}(\vec{ t})\) is a centered multi-dimensional Brownian motion with correlated components as \(u \to \infty\) and \(\vec{ c} \cdot \vec{ t}\) is a component-wise multiplication. We specify conditions which are sufficient to observe no dimension reduction and present exact asymptotics under restrictions - \[\label{assumpptionA} A \gneqq \bf 0, \vec{ \alpha} \Sigma_{\vec{ t}}^{-1} > 0, \quad \vec{ \alpha} > \vec{ 0},\vec{ t }\in [0,1]^d,\] where the studied process \(W_t\) is defined as \[\vec{ W}(\vec{ t}) = A \vec{ B}(\vec{ t})\] with \(\vec{ B}(\vec{ t})\) \(d-\)dimensional Brownian motion with independent coordinates and \(\Sigma_{\vec{ t}}\) is a correlation matrix of \(\vec{ W}(\vec{ t})\). The conditions above enforce positive correlations between components. The above assumptions go in line with observations of real financial market, e.g. in \([6]\) it has been noticed that creating homogeneous groups is a viable strategy for designing the risk models for larger financial portfolios. As mentioned in \([7]\) large companies often show a positive correlation, since their performance is more dependent on the state of the economy as a whole than on the cross-company competition. Additionally, in many sectors a positive correlation between companies occurs because of high dependence of those sectors on external factors and hence the need to model positively correlated portfolios. Similarly, claims for specific kinds of insurance (i.e. weather insurance) can have high positive correlation. We additionally find what is the most likely time of ruin for \(\vec{ W} (\vec{ t})\) and provide upper bounds.

Bibliography

\([1]\) He, H. and Keirstead, W. and Rebholz, J., "Double lookbacks", Mathematical Finance, Vol. 8, No. 3 (July 1998), 201–228

\([2]\) Metzler, A. "On the first passage problem for correlated Brownian motion." Statistics & Probability Letters, vol.80, no.5-6, 2010, pp.277–284.

\([3]\) Shao, J. and Wang, X. "Estimates of the exit probability for two correlated Brownian motions." Advances in Applied Probability, vol.45, no.1, 2013, pp.37–50.

\([4]\) Dȩbicki, K. and Ji, L. and Rolski, T. "Logarithmic asymptotics for probability of component-wise ruin in a two-dimensional Brownian model." Risks, vol.7, no.3, 2019.

\([5]\) Dȩbicki, K. and Hashorva, E. and Krystecki, K. "Finite-time ruin probability for correlated Brownian motions." Scandinavian Actuarial Journal, vol.10, 2021, pp.890–915.

\([6]\) Elton, E. J and Gruber, M. J "Improved forecasting through the design of homogeneous groups." The Journal of Business, vol.44, no.4, 1971, pp.432–450.

\([7]\) Bonanno, G. and Lillo, F. and Mantegna, R. N "High-frequency cross-correlation in a set of stocks." 2001.