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(2)
Covariance Function
Gaussian Process
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On the Variance of the Number of Zeros of a Stationary Gaussian Process
On the Variance of the Number of Zeros of a Stationary Gaussian Process,10.1214/aoms/1177692560,The Annals of Mathematical Statistics,Donald Geman
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On the Variance of the Number of Zeros of a Stationary Gaussian Process
(
Citations: 8
)
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Donald Geman
For a real, stationary
Gaussian process
$X(t)$, it is well known that the mean number of zeros of $X(t)$ in a bounded interval is finite exactly when the
covariance function
$r(t)$ is twice differentiable. Cramer and Leadbetter have shown that the variance of the number of zeros of $X(t)$ in a bounded interval is finite if $(r"(t)  r"(0))/t$ is integrable around the origin. We show that this condition is also necessary. Applying this result, we then answer the question raised by several authors regarding the connection, if any, between the existence of the variance and the existence of continuously differentiable sample paths. We exhibit counterexamples in both directions.
Journal:
The Annals of Mathematical Statistics
, vol. 43, no. 1972, pp. 977982, 1972
DOI:
10.1214/aoms/1177692560
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Citation Context
(2)
...Geman proved in 1972 (see [
54
]) that the condition (2.9) was not only sufficient but also necessary, by showing that if r...
Marie F. Kratz
.
Level crossings and other level functionals of stationary Gaussian pro...
...Combining (16) and (ii) of Lemma 3 allows to conclude that a necessary and sucien t condition to have M2 < 1 is that L 2 L1[0; ]. Thus we nd back Geman’s result ([
7
])...
Marie F. Kratz
.
On the second moment of the number of crossings by a stationary Gaussi...
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Citations
(8)
Counting level crossings by a stochastic process
(
Citations: 1
)
Loren D. Lutes
Journal:
Probabilistic Engineering Mechanics  PROBABILISTIC ENG MECH
, vol. 22, no. 3, pp. 293300, 2007
Level crossings and other level functionals of stationary Gaussian processes
(
Citations: 6
)
Marie F. Kratz
Journal:
Probability Surveys
, no. 2006, pp. 230288, 2006
On the second moment of the number of crossings by a stationary Gaussian process
(
Citations: 5
)
Marie F. Kratz
,
José R. León
Journal:
Annals of Probability  ANN PROBAB
, vol. 34, no. 2006, pp. 16011607, 2006
Central Limit Theorems for Level Functionals of Stationary Gaussian Processes and Fields
(
Citations: 6
)
Marie F. Kratz
,
José R. León
Journal:
Journal of Theoretical Probability  J THEOR PROBABILITY
, vol. 14, no. 3, pp. 639672, 2001
Central Limit Theorems for the Number of Maxima and an Estimator of the Second Spectral Moment of a Stationary Gaussian Process, with Application to Hydroscience
Marie F. Kratz
,
José R. León
Journal:
Extremes
, vol. 3, no. 1, pp. 5786, 2000