Skewed Stable Variables and Processes.

Hardin, Clyde D.

doi:10.21236/ada150549

Cited by 31 publications

(15 citation statements)

References 1 publication

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“…Hereby, the SαS density function with α ∈ (0, 1) can be approximated by scale mixture of Cauchy as well, or say that the SαS distribution are conditionally Cauchy for 0 < α < 1. To extend such a CMM to the region of α ∈ (1, 2), we consider the following theorem proposed by Hardin [30,32].…”

Section: Cgm Modelmentioning

confidence: 99%

Near‐optimal detection with constant false alarm ratio in varying impulsive interference

Sun

Wang³

et al. 2013

IET signal process.

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As an important class of non-Gaussian statistic model, α-stable distribution has received much attention because of its generality to represent impulsive interference. Unfortunately, it does not have a closed-form probability density function (PDF) except for a few cases. For this reason, suboptimal zero-memory non-linearity (ZMNL) function has to be used as an approximation in designing locally optimal detector, such as classical Cauchy and Gaussian-tailed ZMNL (GZMNL). To enhance the performance of detectors, the authors first investigate the approximate PDFs for the symmetric α-stable. In particular, a simplified version of Cauchy-Gaussian mixture (CGM) model, called bi-parameter CGM (BCGM) model is detailed. This BCGM model has a concise closed-form, and hence is more tractable than the classical Gaussian mixture model and CGM model. Then based on the preset false alarm ratio (FAR), the test threshold is adaptively evaluated by using BCGM to maintain a constant FAR. The authors further devise an algebraic-tailed ZMNL (AZMNL) with a simplified form. Simulation results show that the detector with AZMNL outperforms the ones with classical Cauchy and GZMNL, and achieves near-optimal performance in varying impulsive interference.

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Section: Cgm Modelmentioning

confidence: 99%

Near‐optimal detection with constant false alarm ratio in varying impulsive interference

Sun

Wang³

et al. 2013

IET signal process.

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“…It is well known that in Banach spaces of stable type p there holds an inequality between the r th moment of a p-stable measure and the total mass of its spectral measure, and every Banach space is of stable type p , p < 1 (see, e.g., [5]). Namely, as it was shown by Pisier [2,Lemma 5.4]: if X is a p-stable random vector, 0 < p < 1, 0<r<p, and a is its spectral measure then (7) [E\\X\\<]X>< <Cp^\oiSx)]x>p, 2r_1r(l -r/p)ir /0°° zz-''-1 sin2 udu)~x denotes the rth moment of the standard symmetric p-stable random variable on R (for the value of where crpir) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use Cpir) compare [3]). But Remarks.…”

Section: Estimation Of Constants F(p R)mentioning

confidence: 99%

“…random vectors with the distribution a/aiSx), independent of (a,) and (z;). Then the series (3) cp[aiSx)]xfpY/rjl,PVizi i=i is convergent a.s. and has the distribution p.…”

Section: Introduction and Preliminary Factsmentioning

confidence: 99%

Stable measure of a small ball

Lewandowski¹,

Ryznar²,

Żak³

1992

Proc. Amer. Math. Soc.

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Let μ \mu be a symmetric p p -stable measure on a Banach space ( ( E , | | | | ) (E,||||) . We prove that μ { | | x | | > t } ≤ K t \mu \{ ||x|| > t\} \leq Kt , where the constant K K is independent of all properties of μ \mu except for the measure of the unit ball μ { | | x | | > 1 } \mu \{ ||x|| > 1\} .

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“…Note that these two classes of inputs are distinct. (10) When the input X is an SaS moving average and h is a filter with random sign and time shift relative to F, then the Sas output Y is of the form and shows that even though the output Y is not a moving average (in view of the random sign and time shift in (10)), it is nevertheless equal in law with the SaS moving-average process W: (11) Wet) = roo bets) d~(s).…”

Section: Filter Characterizationmentioning

confidence: 99%

“…Throughout this paper the case of symmetric stable inputs is considered. The necessary ingredients to handle the skewed case should be contained in [10], [17].…”

Section: Introductionmentioning

confidence: 99%

Random filters which preserve the stability of random inputs

Cambanis¹

1988

Advances in Applied Probability

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A stationary stable random processes goes through an independently distributed random linear filter. It is shown that when the input is Gaussian or harmonizable stable, then the output is also stable provided the filter&s transfer function has non-random gain. In contrast, when the input is a non-Gaussian stable moving average, then the output is stable provided the filter&s randomness is due only to a random global sign and time shift.

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Skewed Stable Variables and Processes.

Cited by 31 publications

References 1 publication

Near‐optimal detection with constant false alarm ratio in varying impulsive interference

Near‐optimal detection with constant false alarm ratio in varying impulsive interference

Stable measure of a small ball

Random filters which preserve the stability of random inputs

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