Conditions for the uniform convergence in probability of wavelet decompositions for stochastic processes from the space \(\text{Exp}_{\varphi}(\Omega)\). (Ukrainian, English) Zbl 1224.60068

Kozachenko, Yu. V.; Polos’mak, O. V.

Teor. Jmovirn. Mat. Stat. 81, 76-87 (2009); translation in Theory Probab. Math. Stat. 81, 85-99 (2010).

The authors propose conditions for the uniform convergence in probability of expansions of stochastic processes in function systems generated by wavelets. Let \(X=\{X(t),t\in\mathbb R\}\) be a strictly Orlicz stochastic process of exponential type. Let \(\phi=\{\phi(\lambda),\lambda\in\mathbb R\}\) be an \(f\)-wavelet and let \(\psi=\{\psi(\lambda),\lambda\in\mathbb R\}\) be the \(m\)-wavelet corresponding to \(\phi\), let \(\phi_{jk}(x)=2^{j/2}\phi(2^jx-k)\) and \(\psi_{jk}(x)=2^{j/2}\psi(2^jx-k)\) for \(j\in\mathbb Z\), \(k\in\mathbb Z\). The system \(\{\phi_{0k}(x),\psi_{jk}(x),k\in\mathbb Z,j=0,1,\dots\}\) forms a complete orthonormal system in \(L_2(\mathbb R)\) (see, for example, [I. Daubechies, Ten lectures on wavelets. Philadelphia, PA: SIAM (1992; Zbl 0776.42018)]). Define the wavelet decomposition of a strictly Orlicz stochastic process of exponential type \(X=\{X(t),t\in\mathbb R\}\) as the series \[ X(t)=\sum_{k\in\mathbb Z}\xi_{0k}{\phi}_{0k}(t)+ \sum_{j=0}^{\infty}\sum_{k\in\mathbb Z}\eta_{jk}\psi_{jk}(t), \] where \[ \xi_{0k}=\int_{\mathbb R}X(t)\overline{\phi_{0k}(t)}\, dt,\quad \eta_{jk}=\int_{\mathbb R}X(t)\overline{\psi_{jk}(t)}\,dt. \] The authors find sufficient conditions for the uniform convergence of these decompositions in probability to the original stochastic process \(X(t)\). Namely, they find conditions under which \[ P\left\{ \sup_{0\leq t\leq T}| X(t)-X_{n,k_j}(t)| >\varepsilon \right\}\to0,\quad n\to\infty,k_j\to\infty, \] where \[ X_{n,k_j}(t)=\sum_{| k| \leq k_0}\xi_{0k}{\phi}_{0k}(t)+ \sum_{j=0}^{n-1}\sum_{| k| \leq k_j}\eta_{jk}\psi_{jk}(t) \] in a partial sum of the wavelet decomposition of the strictly Orlicz stochastic process of exponential type \(X=\{X(t),t\in\mathbb R\}\) in the series.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G07	General theory of stochastic processes
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

Orlicz stochastic process; wavelet decomposition; uniform convergence in probability

Citations:

Zbl 0776.42018