Maccone second KLT theorem: KLT of all time-rescaled square Brownian motions

  • Claudio Maccone
Part of the Springer Praxis Books book series (PRAXIS)


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].


Brownian Motion Asymptotic Expansion Bessel Function Approximate Expression Asymptotic Expression 
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  1. 1.
    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.Google Scholar
  2. 2.
    A. Papoulis, Signal Analysis, McGraw-Hill, New York, 1977.zbMATHGoogle Scholar
  3. 3.
    C. Maccone, “Eigenfunctions and Energy for Time-Rescaled Gaussian Processes,” Bollettino dell’Unione Matematica Italiana, Series 6, 3-A (1984), 213–219.MathSciNetGoogle Scholar
  4. 4.
    A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental unctions, Vol. 2, McGraw-Hill, New York, 1953.Google Scholar
  5. 5.
    N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972.zbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.International Academy of Astronautics and Istituto Nazionale di AstrofisicaTorino (Turin)Italy

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