I have difficulty in finding references for the following Lemma (284), that I found in a useful lecture of Prof. Cosma Shalizi.
Can anybody provide me some references?
Thank you in advance.
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(Orthogonal Spectral Increments and Weak Stationarity)
The spectral process of a second order process has orthogonal increments if and only if the process is weakly stationary.
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With spectral process it is meant the spectral respresentation of a stochastic process.
http://www.stat.cmu.edu/~cshalizi/754/notes/lecture-22.pdf
You may check the paper by J B. Lawrie in IMA J Appl Math in 2007 and in Proc. Roy. Soc. London Series A i 2009.
I read the papers but the first is about Wiener-Hopf technique, a method for solving systems of integral equations that found wider use in solving two-dimensional partial differential equations. The second is on the solution of a particular wave guide problem. Orthogonality relationships are involved, but not regarding the spectral representation of weakly stationary porcess.
Thus, the hunt is still open.
Thank you Lakhdar, I have just found also the book:
Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications
Harald Cramér, M. Ross Leadbetter
that concludes the hunt.
Thank you all,
S.
Dear Simone,
one of the most celebrated author in the field you are interested in is Akiva Moiseievich Yaglom (a pupil by AN Kolmogorov), whose books cover not only the question of weak sense (wide sense or Hintčin sense) stationarity and its various extensions, but the related correlation and spectral theory and applications:
[1] A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, Dover Publications, 1962.
[2] A. M. YAGLOM, Correlations Theory of Stationary and Related Random
Function I. Basic Results, Springer–Verlag, 1987.
[3] A. M. YAGLOM, Correlations Theory of Stationary and Related Random
Function II. Supplementary Notes and References, Springer–Verlag, 1987.
With regards,
Tibor Pogany
Thanks you, Hamza and Tibor.
I have seen the Note, and I will certainly get the Yaglom's books.
In the meanwhile I have been successful in demonstrating that a renewal rate function of a renewal process was stationary by using orthogonal incrementes of its spectral representation.
Thanks,
S.
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