SpectralBimeasure - crowlogic/arb4j GitHub Wiki
describe the spectral bimeasure of a harmonizable process mathematically precisely rigorously
A harmonizable process is a generalization of stationary processes that allows for representation through spectral measures. To describe the spectral bimeasure rigorously, we need to understand both weakly and strongly harmonizable processes.
Let {X(t), t ∈ G} be a second-order stochastic process with values in a complex space. This process is called harmonizable if it admits a spectral representation of the form:
X(t) = ∫ᴳ e^(i⟨t,u⟩) dZ(u)
where Z(·) is a σ-additive L²(P)-valued measure[3][4].
The spectral bimeasure F of this process is defined by:
F(A,B) = E[Z(A)Z(B)]
for A,B Borel sets[3][4]. This bimeasure has the following properties:
- F(A,·) and F(·,B) are complex measures for every Borel set A and B[1][2]
- F is not necessarily a measure on the product σ-algebra B(G) ⊗ B(G)[1]
- The covariance function can be represented as: R(s,t) = ∫∫ e^(i(s'λ-t'μ)) dF(λ,μ)[2][3]
A process is weakly harmonizable if:
- F is a bimeasure with finite Frechet variation[2][3]
- The correlation function can be represented as shown above
A process is strongly harmonizable (or Loeve harmonizable) if:
- F is a complex-valued measure of finite Vitali variation[2][4]
- The integral representation coincides with the Lebesgue integral[1]
- The spectral bimeasure is actually a measure on B(G) ⊗ B(G)[1][2]
When F is absolutely continuous with respect to Lebesgue measure, we can define the generalized spectral density as:
S(ω,ω') = ∂²F/∂ω∂ω'
This is the Radon-Nikodym derivative of F, and the covariance function can be written as:
k(x,x') = ∫∫ e^(i(ω^T x - ω'^T x')) S(ω,ω') dωdω'[1]
For discrete harmonizable processes, if X(t) has representation:
X(t) = ∑ e^(iλt), t = 0,1,2,...
with E[dY_X(λ₁)dY*_X(λ₂)] = d²F_XX(λ₁,λ₂) = f_XX(λ₁,λ₂)dλ₁dλ₂, then the function f_XX represents the spectral density of the process[5].
The spectral analysis of these processes is more complex than for stationary processes since the spectral mass may not concentrate only on the diagonal line λ = μ[6][7].
Citations: [1] [PDF] Nonstationary multi-output Gaussian processes via harmonizable ... https://proceedings.mlr.press/v151/altamirano22a/altamirano22a.pdf [2] [PDF] c-*/z, - {J5-&A https://ntrs.nasa.gov/api/citations/19890008077/downloads/19890008077.pdf [3] The spectral domain of multivariate harmonizable processes - PNAS https://www.pnas.org/doi/pdf/10.1073/pnas.81.14.4611 [4] 1: Harmonizability and Stochastic Analysis https://www.worldscientific.com/doi/pdf/10.1142/9789811213663_0001 [5] [PDF] A Spectral Estimation of Discrete Harmonizable Process https://www.naturalspublishing.com/download.asp?ArtcID=25597 [6] Spectral analysis for harmonizable processes - Project Euclid https://projecteuclid.org/journals/annals-of-statistics/volume-30/issue-1/Spectral-analysis-for-harmonizable-processes/10.1214/aos/1015362193.pdf [7] [PDF] Nonstationary multi-output Gaussian processes via harmonizable ... https://repositorio.uchile.cl/bitstream/handle/2250/182343/Nonstationary-multi-output-gaussian-processes-via-harmonizable-spectral-mixtures.pdf?sequence=1 [8] Line spectral analysis for harmonizable processes - PMC https://pmc.ncbi.nlm.nih.gov/articles/PMC20166/ [9] [PDF] Spectral Analysis for harmonizable processes - UCSD Math https://mathweb.ucsd.edu/~mrosenbl/papers/Spectral_Analysis_for_Harmonizable_Processes.pdf [10] Conditions for the completeness of the spectral domain of a ... https://www.sciencedirect.com/science/article/pii/S030441499700080X/pdf?md5=ce3775d816daf38b6a8f23bc0684b7d1&pid=1-s2.0-S030441499700080X-main.pdf [11] A Spectral Estimation of Discrete Harmonizable Process https://digitalcommons.aaru.edu.jo/isl/vol12/iss2/1/ [12] An Example of a Harmonizable Process Whose Spectral Domain Is ... https://www.jstor.org/stable/4616207 [13] Harmonizable Nonstationary Processes - SIAM.org https://epubs.siam.org/doi/10.1137/22M1544580 [14] Spectral domains of vector harmonizable processes - ScienceDirect https://www.sciencedirect.com/science/article/abs/pii/S0378375801001252 [15] Spectral Analysis for Harmonizable Processes - jstor https://www.jstor.org/stable/2700011 [16] [PDF] SOME PROPERTIES OF HARMONIZABLE PROCESSES - Marc ... https://cirrus18.com/articles/prop.pdf [17] On a class of asymptotically stationary harmonizable processes https://www.sciencedirect.com/science/article/pii/0047259X87900893 [18] [PDF] Harmonizable, Cramer, and Karhunen Classes of Processes. - DTIC https://apps.dtic.mil/sti/tr/pdf/ADA144984.pdf [19] [PDF] Harmonizable Processes - DTIC https://apps.dtic.mil/sti/tr/pdf/ADA093302.pdf [20] Harmonic measure in the presence of a spectral gap - Project Euclid https://projecteuclid.org/journals/annales-de-linstitut-henri-poincare-probabilites-et-statistiques/volume-52/issue-3/Harmonic-measure-in-the-presence-of-a-spectral-gap/10.1214/15-AIHP670.pdf [21] A First Course in Spectral Theory - American Mathematical Society https://www.ams.org/books/gsm/226/gsm226-endmatter.pdf [22] [PDF] Bimeasures and Sampling Theorems for Weakly Harmonizable ... https://apps.dtic.mil/sti/tr/pdf/ADA120937.pdf [23] [PDF] Spectral theory in Hilbert spaces (ETH Zürich, FS 09) E. Kowalski https://people.math.ethz.ch/~kowalski/spectral-theory.pdf [24] [PDF] Preliminaries In the following work, let (Ω,Σ,P) be the underl - EMIS https://www.emis.de/journals/PM/57f1/pm57f105.pdf [25] Measure (mathematics) - Wikipedia https://en.wikipedia.org/wiki/Measure_(mathematics) [26] [PDF] Characterizations of harmonizable fields - Sci-Hub https://dacemirror.sci-hub.se/journal-article/e42b0f933bae2cfda4d539a9989aa8f9/rao2005.pdf [27] [PDF] A GUIDE TO SPECTRAL THEORY https://perso.univ-rennes1.fr/christophe.cheverry/spectral-theory.pdf [28] Advances on Theoretical and Methodological Aspects of Probability ... https://api-uat.taylorfrancis.com/content/chapters/edit/download?identifierName=doi&identifierValue=10.1201%2F9780203493205-2&type=chapterpdf [29] Time-frequency characterization of harmonizable random processes https://munin.uit.no/handle/10037/5925 [30] [PDF] On a Class of Asymptotically Stationary Harmonizable Processes https://core.ac.uk/download/pdf/82720645.pdf