Negative-order moments for \( L^p\)-functionals of Wiener processes: exact asymptotics (Q2901883)

The author presents theorems on the exact asymptotic behaviour as \(T \to \infty\) of the integral NEWLINE\[NEWLINE\operatorname{E}\left[ \frac{1}{T}\int_0^T |\eta (t)|^p\,dt\right]^{-T},\quad p>0NEWLINE\]NEWLINE for the two cases of the Wiener process and the Brownian bridge as well as their conditional versions.NEWLINENEWLINEThere are four types of problems related to this paper.NEWLINENEWLINE(i) The problem of moments of large positive order related to understanding the two limits NEWLINE\[NEWLINE\lim_{T\to \infty} \operatorname{E}\left[ \int_0^S |\eta (t)|^p\,dt\right]^{T}\quad\text{and}\quad\lim_{m\to \infty} \operatorname{E}\left[ \int_0^S (\eta (t))^{\frac{2k-1}{2l-1}}\,dt\right]^{2m}NEWLINE\]NEWLINE for \(S>0,\;p>0,\;k, l, m = 1, 2, \dots\).NEWLINENEWLINE(ii) The problem of moments of large negative order related to understanding the limit NEWLINE\[NEWLINE\lim_{T\to \infty} \operatorname{E}\left[ \int_0^S |\eta (t)|^p\,dt\right]^{-T}.NEWLINE\]NEWLINENEWLINENEWLINE(iii) The problem of moments of ergodic with large positive exponent related to the calculation of NEWLINE\[NEWLINE\lim_{T\to \infty} \operatorname{E}\left[ \frac{1}{T}\int_0^T |\eta (t)|^p\,dt\right]^{T}.NEWLINE\]NEWLINENEWLINENEWLINE(iv) The problem of moments of ergodic with large negative exponent related to the calculation of NEWLINE\[NEWLINE\lim_{T\to \infty} \operatorname{E}\left[ \frac{1}{T}\int_0^T |\eta (t)|^p\,dt\right]^{-T}.NEWLINE\]NEWLINENEWLINENEWLINEThe author solved (i) and (iii) for the Wiener process and the Brownian bridge and (i) for the Bogolyubov Gaussian process in [``The moments of \(L^p\)-functionals of Gaussian processes: exact asymptotics'', Probl. Inf. Transm. (to appear)].NEWLINENEWLINEThe goal paper under review is to provide answers to (ii) and (iv) for the Wiener process and Brownian bridges as well as for conditional versions of a Wiener Markov process. Theorem 1.1 shows that a limit of type (iv) is given by the normalized positive eigenfunction \(y_0\) with the smallest positive eigenvalue \(k_0\), which characterizes the exact asymptotic behaviour of the distribution of \(L^p\)-functionals of the Wiener process in the small deviation case and the large negative exponent case. The constants in Theorem 1.1 are computed by using Airy functions. For \(p=2\), the special pair \((k_0, y_0)\) are computed explicitly in Corollary 1.2. Then the answer (given in Proposition 1.1) to (ii) can be reduced to self-similarity of the Wiener process starting at zero from Theorem 1.1. A similar statement for the Brownian bridge is given with Theorem 1.2, as well as the special case of \(p=2\) (Corollary 1.2).NEWLINENEWLINESection 2 starts by building a relation between (iv) and the theory of large deviations for occupation times. It proceeds by relating the Wiener process to an \(m\)-symmetric Markov process so that one can apply the effective theory of large deviations for \(m\)-symmetric Markov processes, and uses \textit{S. Kusuoka} and \textit{Y. Tamura}'s 1991 results on a ``Precise estimate for large deviation of Donsker-Varadhan type'' [J. Fac. Sci., Univ. Tokyo, Sect. I A 38, No. 3, 533--565 (1991; Zbl 0752.60026)] to derive Theorem 2.1 (exact asymptotic behaviour of Wiener integrals of Laplace type).NEWLINENEWLINEThe proof of Theorem 1.1 in Section 3 (the main result of the paper) proceeds (a) by the proof of Theorem 2.1 in Subsection 3.1 (many arguments are verbatim or modified ones from [loc. cit.]); (b) by identifying the explicit terms in Theorem 2.1 for the Wiener process and the Brownian bridge as well as for the case of \(p=2\) in Subsection 3.2.NEWLINENEWLINEThe paper certainly contributes to further the understanding of other stochastic processes, and it would be nice to have a detailed proof of Theorem 1.1, since the present paper refers to other articles (some are not easy to access or available at all to the reader).