Zero Crossing Probabilities for Gaussian Stationary Processes

Newell, G. F.; Rosenblatt, Murray

doi:10.1214/aoms/1177704363

Cited by 65 publications

(82 citation statements)

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“…Early papers on gap probabilities are by Longuet-Higgins [9] and Newell and Rosenblatt [14]. Newell and Rosenblatt [14] obtain a number of bounds for the gap probability P{X t > 0 for t ∈ [0, T ]} for a stationary Gaussian process on R. They showed that if the covariance of X 0 and X m goes to zero as m → ∞, then the persistence probability H X (Q N ) decays faster than any polynomial in N and if Cov(X 0 , X m ) is also summable, then they showed that H X (N ) ≤ e −cN .…”

Section: Brief Review Of Past Resultsmentioning

confidence: 99%

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Persistence probabilities in centered, stationary, Gaussian processes in discrete time

Krishna

Krishnapur

2016

Indian J Pure Appl Math

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THE PROBLEM AND OUR RESULTSLet X = (X m ) m∈Z d be a centered, stationary Gaussian process on Z d . This means that for any k ≥ 1 and any m 0 , . . . , m k ∈ Z d , the vector (X mj +m0 ) 1≤j≤k has a multivariate Gaussian distribution with zero mean and a covariance matrix that does not depend on m 0 . For basics on Gaussian processes, consult for example, the book by Adler [1].For a subset A ⊆ Z d , we define the persistence probability (also called gap probability or hole probability) of X in A asIn particular, one may be interested inThis paper is exclusively about getting bounds on the persistence probability under some additional conditions on the Gaussian process.A stationary Gaussian process on Z d is uniquely described by its covariance kernel Cov(X m , X n ) = Cov(X 0 , X m−n ). Further, there exists a unique finite Borel measure µ on T d that is symmetric about the origin (i.e., µ(I) = µ(−I) for any Borel set I ⊆ T d ) such that Cov(X 0 , X m ) =μ(m) whereμ(m) = T d e i m,t dµ(t) with the usual notation for the inner product m, t = m 1 t 1 + . . . + m d t d . The measure µ is called the spectral measure of the process X. Write dµ(t) = b(t)dλ(t) + dµ s (t) where µ s is singular to Lebesgue measure and b ∈ L 1 (T d , λ) is non-negative. In all the results of this paper, it will be assumed that b is not identically zero. In other words, the spectral measure is not singular. Lastly, for a subset A ⊆ Z d , we denote the covariance matrix of (X m ) m∈A by Σ A := (μ(j − k)) j,k∈A and the cardinality of A by |A|.These notations will be maintained throughout the paper without further mention. In addition, there will appear many constants denoted by C, c, γ etc. Unless otherwise mentioned, the constants depend on the given process X (or equivalently, on the spectral measure µ).We now state our results and then give an overview of past results in the literature in Section 2. Our first theorem has already been proved by N. Feldheim and O. Feldheim [8] and but we explain in Section 2 why we include it here nevertheless.

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Section: Brief Review Of Past Resultsmentioning

confidence: 99%

“…This example is one among a larger class of time series considered by Majumdar and Dhar [11]. Newell and Rosenblatt [14] also remark in their paper that if the covariance is not positive, then the gap probability can decay faster than exponential, but they do not give an example. Example 6.…”

Section: Exponentialmentioning

confidence: 98%

Persistence probabilities in centered, stationary, Gaussian processes in discrete time

Krishna

Krishnapur

2016

Indian J Pure Appl Math

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“…The distribution of return times t r in the presence of LTM is approximately given by a stretched exponential, p∼ exp(−t γ r ), where the exponent is assumed to be identical to the correlation exponent γ . The stretched exponential is motivated by the study of Newell and Rosenblatt (1962) who derived an upper bound for the probability of no zero crossings in power-law correlated Gaussian processes. Olla (2007) applied an -expansion for γ =1− and obtained a stretched exponential distribution with exponent γ .…”

Section: Introductionmentioning

confidence: 99%

Extreme event return times in long-term memory processes near 1/<i>f</i>

Blender

Fraedrich

Sienz

2008

Nonlin. Processes Geophys.

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Abstract. The distribution of extreme event return times and their correlations are analyzed in observed and simulated long-term memory (LTM) time series with 1/f power spectra. The analysis is based on tropical temperature and mixing ratio (specific humidity) time series from TOGA COARE with 1 min resolution and an approximate 1/f power spectrum. Extreme events are determined by PeakOver-Threshold (POT) crossing. The Weibull distribution represents a reasonable fit to the return time distributions while the power-law predicted by the stretched exponential for 1/f deviates considerably.For a comparison and an analysis of the return time predictability, a very long simulated time series with an approximate 1/f spectrum is produced by a fractionally differenced (FD) process. This simulated data confirms the Weibull distribution (a power law can be excluded). The return time sequences show distinctly weaker long-term correlations than the original time series (correlation exponentγ ≈0.56).

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“…The central idea in [8] is that threshold crossings will be less likely for processes with longer correlations. Comparing with processes with known return time distribution (typically, a superposition of an Ornstein-Uhlembeck process and a stochastic variable) Newell and Roseblatt were able to prove that the no zero-crossing probability for y is bounded from above and from below by stretched exponentials.…”

Section: Introductionmentioning

confidence: 99%

“…In principle, the approach in [8] could be extended to the case of a threshold different from zero, allowing to conclude that stretched exponential is the correct asymptotic scaling of the return probability for large values of its argument. This leaves open, however, some important questions.…”

Section: Introductionmentioning

confidence: 99%

Return times for stochastic processes with power-law scaling

Olla

2007

Phys. Rev. E

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An analytical study of the return time distribution of extreme events for stochastic processes with power-law correlation has been carried on. The calculation is based on an ǫ-expansion in the correlation exponent: C(t) = |t| −1+ǫ . The fixed point of the theory is associated with stretched exponential scaling of the distribution; analytical expressions, valid in the pre-asymptotic regime, have been provided. Also the permanence time distribution appears to be characterized by stretched exponential scaling. The conditions for application of the theory to non-Gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed.

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Zero Crossing Probabilities for Gaussian Stationary Processes

Cited by 65 publications

References 4 publications

Persistence probabilities in centered, stationary, Gaussian processes in discrete time

Persistence probabilities in centered, stationary, Gaussian processes in discrete time

Extreme event return times in long-term memory processes near 1/<i>f</i>

Return times for stochastic processes with power-law scaling

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Zero Crossing Probabilities for Gaussian Stationary Processes

Cited by 65 publications

References 4 publications

Persistence probabilities in centered, stationary, Gaussian processes in discrete time

Persistence probabilities in centered, stationary, Gaussian processes in discrete time

Extreme event return times in long-term memory processes near 1/&lt;i&gt;f&lt;/i&gt;

Return times for stochastic processes with power-law scaling

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Extreme event return times in long-term memory processes near 1/<i>f</i>