• Bernoulli
  • Volume 25, Number 4A (2019), 2949-2981.

On logarithmically optimal exact simulation of max-stable and related random fields on a compact set

Zhipeng Liu, Jose H. Blanchet, A.B. Dieker, and Thomas Mikosch

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We consider the random field \begin{equation*}M(t)=\mathop{\mathrm{sup}}_{n\geq1}\{-\log A_{n}+X_{n}(t)\},\qquad t\in T,\end{equation*} for a set $T\subset\mathbb{R}^{m}$, where $(X_{n})$ is an i.i.d. sequence of centered Gaussian random fields on $T$ and $0<A_{1}<A_{2}<\cdots$ are the arrivals of a general renewal process on $(0,\infty)$, independent of $(X_{n})$. In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that one needs $c(d)=c(\{t_{1},\ldots,t_{d}\})$ function evaluations to sample $X_{n}$ at $d$ locations $t_{1},\ldots,t_{d}\in T$. We provide an algorithm which samples $M(t_{1}),\ldots,M(t_{d})$ with complexity $O(c(d)^{1+o(1)})$ as measured in the $L_{p}$ norm sense for any $p\ge1$. Moreover, if $X_{n}$ has an a.s. converging series representation, then $M$ can be a.s. approximated with error $\delta$ uniformly over $T$ and with complexity $O(1/(\delta\log(1/\delta))^{1/\alpha})$, where $\alpha$ relates to the Hölder continuity exponent of the process $X_{n}$ (so, if $X_{n}$ is Brownian motion, $\alpha=1/2$).

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Bernoulli, Volume 25, Number 4A (2019), 2949-2981.

Received: September 2016
Revised: September 2018
First available in Project Euclid: 13 September 2019

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Brown–Resnick process exact simulation Gaussian field max-stable random fields record-breaking


Liu, Zhipeng; Blanchet, Jose H.; Dieker, A.B.; Mikosch, Thomas. On logarithmically optimal exact simulation of max-stable and related random fields on a compact set. Bernoulli 25 (2019), no. 4A, 2949--2981. doi:10.3150/18-BEJ1076.

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  • [1] Adler, R.J. and Taylor, J.E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. New York: Springer.
  • [2] Asmussen, S. (2003). Applied Probability and Queues: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 51. New York: Springer.
  • [3] Asmussen, S., Glynn, P. and Pitman, J. (1995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Ann. Appl. Probab. 5 875–896.
  • [4] Asmussen, S. and Glynn, P.W. (2007). Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability 57. New York: Springer.
  • [5] Ayache, A. and Taqqu, M.S. (2003). Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl. 9 451–471.
  • [6] Blanchet, J. and Chen, X. (2015). Steady-state simulation of reflected Brownian motion and related stochastic networks. Ann. Appl. Probab. 25 3209–3250.
  • [7] Blanchet, J., Chen, X. and Dong, J. (2017). $\varepsilon$-strong simulation for multidimensional stochastic differential equations via rough path analysis. Ann. Appl. Probab. 27 275–336.
  • [8] Blanchet, J. and Wallwater, A. (2015). Exact sampling of stationary and time-reversed queues. ACM Trans. Model. Comput. Simul. 25 Art. 26.
  • [9] Blanchet, J.H. and Sigman, K. (2011). On exact sampling of stochastic perpetuities. J. Appl. Probab. 48A 165–182.
  • [10] Brown, B.M. and Resnick, S.I. (1977). Extreme values of independent stochastic processes. J. Appl. Probab. 14 732–739.
  • [11] de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab. 12 1194–1204.
  • [12] de Haan, L. and Zhou, C. (2008). On extreme value analysis of a spatial process. REVSTAT 6 71–81.
  • [13] Dieker, A.B. and Mikosch, T. (2015). Exact simulation of Brown–Resnick random fields at a finite number of locations. Extremes 18 301–314.
  • [14] Dombry, C., Engelke, S. and Oesting, M. (2016). Exact simulation of max-stable processes. Biometrika 103 303–317. Available at arXiv:1506.04430.
  • [15] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Berlin: Springer.
  • [16] Gut, A. (2009). Stopped Random Walks: Limit Theorems and Applications, 2nd ed. Springer Series in Operations Research and Financial Engineering. New York: Springer.
  • [17] Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37 2042–2065.
  • [18] Kenealy, B. (2013, August 11). New York’s MTA buys $200 million cat bond to avoid storm surge losses. Bus. Insur. Available at
  • [19] Oesting, M., Schlather, M. and Zhou, C. (2018). Exact and fast simulation of max-stable processes on a compact set using the normalized spectral representation. Bernoulli 24 1497–1530.
  • [20] Pollock, M., Johansen, A.M. and Roberts, G.O. (2016). On the exact and $\varepsilon$-strong simulation of (jump) diffusions. Bernoulli 22 794–856.
  • [21] Schilling, R.L. and Partzsch, L. (2012). Brownian Motion: An Introduction to Stochastic Processes. Berlin: de Gruyter. With a chapter on simulation by Björn Böttcher.
  • [22] Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5 33–44.
  • [23] Smith, R.L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript.
  • [24] Steele, J.M. (2001). Stochastic Calculus and Financial Applications. Applications of Mathematics (New York) 45. New York: Springer.
  • [25] Thibaud, E., Aalto, J., Cooley, D.S., Davison, A.C. and Heikkinen, J. (2016). Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures. Ann. Appl. Stat. 10 2303–2324. Available at arXiv:1506.07836.