Abstract
Various methods to derive new formulas for the Laplace transforms of some quadratic forms of Gaussian sequences are discussed. In the general setting, an approach based on the resolution of an appropriate auxiliary filtering problem is developed; it leads to a formula in terms of the solutions of Volterra‐type recursions describing characteristics of the corresponding optimal filter. In the case of Gauss‐Markov sequences, where the previous equations reduce to ordinary forward recursive equations, an alternative approach prices another formula; it involves the solution of a backward recursive equation. Comparing the different formulas for the Laplace transforms, various relationships between the corresponding entries are identified. In particular, relationships between the solutions of matched forward and backward Riccati equations are thus proved probabilistically; they are proved again directly. In various specific cases, a further analysis of the concerned equations lead to completely explicit formulas for the Laplace transform.
Highlights
Quadratic functionals of Gaussian processes have attracted a great deal of interest over the past decades
It is worth mentioning that identities (6) and (7) say that one may compute the determinant and quadratic form appearing in the left-hand sides by applying procedures (4) and (5)
An an immediate consequence of the filtering approach developed in Section 2, we get a second formula for the Ltqf
Summary
Quadratic functionals of Gaussian processes have attracted a great deal of interest over the past decades. We concentrate on Laplace transforms of quadratic forms (Ltqf, for short) of Gaussian sequences. The auxiliary results, which are themselves of independent interest, are investigated in Appendices A and B: the filtering problem introduced in Section 2 is solved and identities connected with the Riccati equations are proved again directly
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have