Abstract
A surprising feature of the KL expansion obtained in Chapter 22 is that the same analytical solution valid for the X(t) process can be carried over to the X2(t) process. In other words, to keep within the easy framework of standard Brownian motion B(t), if we know the KL expansion of B(t), then we may also find the KL expansion of B2(t). The latter will actually be computed at the end of the present chapter, but, as mentioned above, the general proof is valid for any time-rescaled Brownian motion X2(t). The results proved in this Appendix were discovered by the author in 1988 and published in [1].
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References
C. Maccone, “The Karhunen–Loève Expansion of the Square of a Time-Rescaled Gaussian Process,” Bollettino dell’Unione Matematica Italiana, Series 7, 2-A (1988), 21–229.
A. Papoulis, Signal Analysis, McGraw-Hill, New York, 1977.
C. Maccone, “Eigenfunctions and Energy for Time-Rescaled Gaussian Processes,” Bollettino dell’Unione Matematica Italiana, Series 6, 3-A (1984), 213–219.
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental unctions, Vol. 2, McGraw-Hill, New York, 1953.
N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972.
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Maccone, C. (2012). Maccone second KLT theorem: KLT of all time-rescaled square Brownian motions. In: Mathematical SETI. Springer Praxis Books(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27437-4_24
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DOI: https://doi.org/10.1007/978-3-642-27437-4_24
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