Laws of iterated logarithm for stochastic integrals of generalized sub-Gaussian processes. (English) Zbl 1120.60023

Castellucci, A.; Antonini, R. Giuliano

Teor. Jmovirn. Mat. Stat. 73, 43-51 (2005) and Theory Probab. Math. Stat., Vol 73, 47-56 (2006).

The class of \(\varphi\)-sub-Gaussian random variables \(X\) was introduced and studied by V. V. Buldygin and Yu. V. Kozachenko [see, for example, “Metric characterization of random variables and random processes” (1998; Zbl 0933.60031)], where the importance of the Young-Fenchel transform \(\varphi^\ast\) of \(\varphi\) for the tail probabilities \(P(X>x)\) is shown. The authors of this paper show that the Young-Fenchel transform plays a similar role in the behaviour of \(\varphi\)-sub-Gaussian martingales \((M_t)_{t>o}\) in the neighbourhood of \(0\). It is shown that \((M_t)_{t>0}\) behaves in some sense like the function \(t\to(\varphi^\ast)^{-1}(\log\log(1/t))\) that reminds us of the classical law of the iterated logarithm. Applications are given to the stochastic integral of a process \((Y_t)\) such that its \(L^{2k}\) norms \(\| Y_t\| _{L^{2k}}\) are well controlled and to the double stochastic integral of a process \((X_t)\) with well controlled \(L^{2k}\) norms \(\| X_t\| _{L^{2k}}\) with respect to two independent Brownian motions.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60F15	Strong limit theorems
60G44	Martingales with continuous parameter
60H05	Stochastic integrals

Keywords:

continuous time martingale; generalized sub-Gaussian process; iterated logarithm law; Brownian motion; double stochastic integral; Lévy area process

Citations:

Zbl 0998.60503; Zbl 0933.60031