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- Lindgren, Georg
(författare)
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A note on the asymptotic independence of high level crossings for dependent Gaussian processes
- 1974
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798. ; 2:3, s. 535-539
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that the numbers of high level crossings by p dependent stationary Gaussian processes have asymptotically independent Poisson distributions if the observation interval and the height of the level increase in a coordinated way. The processes may be highly correlated, even linearly dependent.
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| 3. |
- Lindgren, Georg
(författare)
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Discrete wave-analysis of continuous stochastic processes
- 1973
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 1:1, s. 83-105
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Tidskriftsartikel (refereegranskat)abstract
- he behaviour of a continuous-time stochastic process in the neighbourhood of zero-crossings and local maxima is compared with the behaviour of a discrete sampled version of the same process.For regular processes, with finite crossing-rate or finite rate of local extremes, the behaviour of the sampled version approaches that of the continuous one as the sampling interval tends to zero. Especially the zero-crossing distance and the wave-length (i.e., the time from a local maximum to the next minimum) have asymptotically the same distributions in the discrete and the continuous case. Three numerical illustrations show that there is a good agreement even for rather big sampling intervals.For non-regular processes, with infinite crossing-rate, the sampling procedure can yield useful results. An example is given in which a small irregular disturbance is superposed over a regular process. The structure of the regular process is easily observable with a moderate sampling interval, but is completely hidden with a small interval.
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| 5. |
- Lindgren, Georg
(författare)
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Functional limits of empirical distributions in crossing theory
- 1977
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 5:2, s. 143-149
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Tidskriftsartikel (refereegranskat)abstract
- We present a functional limit theorem for the empirical level-crossing behaviour of a stationary Gaussian process. This leads to the well-known Slepian model process for a Gaussian process after an upcrossing of a prescribed level as a weak limit in C-space for an empirically defined finite set of functions.We also stress the importance of choosing a suitable topology by giving some natural examples of continuous and non-continuous functionals.
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| 6. |
- Lindgren, Georg
(författare)
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Local maxima of Gaussian fields
- 1972
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Ingår i: Arkiv för Matematik. - 0004-2080. ; 10:1-2, s. 195-218
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Tidskriftsartikel (refereegranskat)abstract
- The structure of a stationary Gaussian process near a local maximum with a prescribed heightu has been explored in several papers by the present author, see [5]–[7], which include results for moderateu as well as foru→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ R n}, with mean zero and the covariance functionr(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type. In Section 1 it is shown that if ξ has a local maximum with heightu at0 then ξ(t) can be expressed as $$\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$$ WhereA(t) andb(t) are certain functions, θu is a random vector, and Δ(t) is a non-homogeneous Gaussian field. Actually ξu(t) is the old process ξ(t) conditioned in the horizontal window sense to have a local maximum with heightu fort=0; see [4] for terminology. In Section 2 we examine the process ξu(t) asu→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial int 1…,t n with random coefficients. This result is quite analogous with the one-dimensional case. In Section 3 we study the locations of the local minima of ξu(t) asu → ∞. In the non-isotropic caser(t) may have a local minimum at some pointt 0. Then it is shown in 3.2 that ξu(t) will have a local minimum at some point τu neart 0, and that τu-t 0 after a normalization is asymptoticallyn-variate normal asu→∞. This is in accordance with the one-dimensional case.
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| 8. |
- Lindgren, Georg
(författare)
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Prediction from a random time point
- 1975
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Ingår i: Annals of Probability. - 0091-1798. ; 3:3, s. 412-423
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Tidskriftsartikel (refereegranskat)abstract
- In prediction (Wiener-, Kalman-) of a random normal process $\{X(t), t \in R\}$ it is normally required that the time $t_0$ from which prediction is made does not depend on the values of the process. If prediction is made only from time points at which the process takes a certain value $u,$ given a priori, ("prediction under panic"), the Wiener-prediction is not necessarily optimal; optimal should then mean best in the long run, for each single realization. The main theorem in this paper shows that when predicting only from upcrossing zeros $t_\nu$, the Wiener-prediction gives optimal prediction of $X(t_\nu + t)$ as $t_\nu$ runs through the set of zero upcrossings, if and only if the derivative $X'(t_\nu)$ at the crossing points is observed. The paper also gives the conditional distribution from which the optimal predictor can be computed.
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