Correlated additive and multiplicative (CAM) noise processes are well-established as general “null hypothesis” models of non-Gaussian variability in atmospheric and oceanic quantities. In this study, analytic expressions for the bispectral density (which partitions the third statistical moment into triad frequency interactions in a manner analogous to the partitioning of variance by the spectral density) are developed for discrete and continuous-time CAM processes. It is then demonstrated that under lowpass filtering, while the absolute skewness of a discrete-time CAM process may increase or decrease with decreasing cutoff frequency, the absolute skewness of continuous-time CAM processes decreases monotonically. This second result provides a test to assess the degree to which an observed time series is consistent with continuous-time CAM dynamics.

1.
P.
Sura
and
A.
Hannachi
, “
Perspectives of non-Gaussianity in atmospheric synoptic and low-frequency variability
,”
J. Clim.
28
,
5091
5114
(
2015
).
2.
P. D.
Sardeshmukh
and
P.
Sura
, “
Reconciling non-Gaussian climate statistics with linear dynamics
,”
J. Clim.
22
,
1193
1207
(
2009
).
3.
P.
Sura
, “
A general perspective of extreme events in weather and climate
,”
Atmos. Res.
101
,
1
21
(
2011
).
4.
C.
Penland
and
P. D.
Sardeshmukh
, “
Alternative interpretations of power-law distributions found in nature
,”
Chaos
22
,
023119
(
2012
).
5.
P. D.
Sardeshmukh
and
C.
Penland
, “
Understanding the distinctively skewed and heavy tailed character of atmospheric and oceanic probability distributions
,”
Chaos
25
,
036410
(
2015
).
6.
P. D.
Sardeshmukh
,
G. P.
Compo
, and
C.
Penland
, “
Need for caution in interpreting extreme weather statistics
,”
J. Clim.
28
,
9166
9187
(
2015
).
7.
D.
Müller
, “
Bispectra of sea-surface temperature anomalies
,”
J. Phys. Ocean.
17
,
26
36
(
1987
).
8.
J. M.
Gairing
,
M. A.
Högele
,
T.
Kosenkova
, and
A. H.
Monahan
, “
How close are time series to power tail Lévy diffusions?
,”
Chaos
27
,
073112
(
2017
).
9.
W. F.
Thompson
,
R. A.
Kuske
, and
A. H.
Monahan
, “
Reduced α-stable dynamics for multiple time scale systems forced with correlated additive and multiplicative Gaussian white noise
,”
Chaos
27
,
113105
(
2017
).
10.
C.
Proistosescu
,
A.
Rhines
, and
P.
Huybers
, “
Identification and interpretation of non-normality in atmospheric time series
,”
Geophys. Res. Lett.
43
,
5425
5434
, https://doi.org/10.1002/2016GL068880 (
2016
).
11.
A. H.
Monahan
, “
Temporal filtering enhances the skewness of sea surface winds
,”
J. Clim.
31
,
5695
5706
(
2018
).
12.
T. S.
Rao
and
M. M.
Gabr
,
An Introduction to Bispectral Analysis and Bilinear Time Series Models
(
Springer Verlag
,
New York
,
1984
), p. 280.
13.
D. B.
Percival
and
A. T.
Walden
,
Spectral Analysis for Physical Applications
(
Cambridge University Press
,
Cambridge, UK
,
1993
), p. 612.
14.
A. M.
Yaglom
,
An Introduction to the Theory of Stationary Random Functions
(
Martino Fine Books
,
2014
), p. 250.
15.
S.
Elgar
, “
Relationships involving third moments and bispectra of a harmonic process
,”
IEEE Trans. Acoust. Speech. Sig. Proc.
ASSP-35
,
1725
1726
(
1987
).
16.
C. W. J.
Granger
and
A. P.
Andersen
,
An Introduction to Bilinear Time Series Models
(
Vandenhoeck & Ruprecht
,
1978
), p. 94.
17.
Y.
Birkelund
, “
Statistical signal processing with higher order spectra: Non-linear signal and system analysis
,” Ph.D. thesis (University of Tromsø, 2002).
18.
Y.
Birkelund
,
A.
Hanssen
, and
E. J.
Powers
, “
Multitaper estimators of polyspectra
,”
Signal Process.
83
,
545
559
(
2003
).
19.
J.
Theiler
,
S.
Eubank
,
A.
Longtin
,
B.
Galdrikian
, and
J. D.
Farmer
, “
Testing for nonlinearity in time series: The method of surrogate data
,”
Physica D
58
,
77
94
(
1992
).
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