The authors investigate relations between the small deviation problem for a symmetric \(\alpha\)-stable random vector in Banach space and the metric entropy properties of the operator generating this random vector. Let \(E\) be a Banach space with the norm \(\|\cdot\|_E\) and let \(E'\) be the (topological) dual space with the weak-*-topology and the corresponding \(\sigma\)-field \({\mathcal B}_{\sigma}(E')\). A measurable with respect to this \(\sigma\)-field \(E'\)-valued random vector on \((\Omega,\mathbb P)\) is said to be symmetric \(\alpha\)-stable (\(S\alpha S\)) for some \(\alpha\in (0, 2]\) if there exist a measure space \((S,\sigma)\) and a bounded linear operator \(u:E\to L_{\alpha}(S,\sigma)\) (called generator of \(X\)) such that \({\mathbb E}e^{i<z,X>}=e^{-\|u\|_{\alpha}^{\alpha}}\), \(z\in E\). A symmetric 2-stable vector is centered Gaussian. In this case, there exist tight relations between the degree of compactness of \(u:E\to L_2\) and small deviation properties of the generated random vector \(X\). The small deviation function of an \(E'\)-valued random vector \(X\) is introduced as \(\varphi(X,\varepsilon):=-\log{\mathbb P}(\|X\|_{E'}<\varepsilon)\) To measure the degree of compactness of the corresponding operator \(u\) the dyadic entropy numbers are used defined as follows: if \(u\) is a bounded linear operator between the Banach spaces \(E\) and \(F\), then \[ e_n(u):=\inf\{ \varepsilon>0|\exists y_1,\dots,y_{2^{n-1}}\in F,\forall z\in E,\|z\|\leq1, \exists i\leq 2^{n-1},\|u(z)-y_i\|\leq\varepsilon\} \] Their behaviour as \(n=\to\infty\) describes the degree of compactness of \(u\). Let us establish some more notation. We write \(f\precsim g\) if \(\lim\sup(f/g)<\infty\), the equivalence \(f\approx g\) means that we have both \(f\precsim g\) and \(g\precsim f\), \(f\lesssim g\) indicate that \(\lim\sup(f/g)<1\), the strong equivalence \(f\sim g\) means that \(\lim(f/g)=1\). Using this notation, we can now state the relations between properties of \(X\) and the generating operator \(u\) in the Gaussian case (see J. Kuelbs and W. V. Li [J. Funct. Anal. 116, No.1, 133–157 (1993; Zbl 0799.46053)]; W. V. Li and W. Linde [Ann. Probab. 27, No.3, 1556–1578 (1999; Zbl 0983.60026)]).
Proposition 1. Let \(X\) be an \(E'\)-valued Gaussian vector generated by the operator \(u:E\to\ell_2\). Let \(\tau>0\) and let \(L\) be a slowly varying function at infinity such that \(L(t)\approx L(t^p)\) for all \(p>0\). The following implications then hold:
(a) we have \[ e_n(u)\succsim n^{-1/2-1/\tau}L(u)\quad \Leftrightarrow \quad \varphi(X,\varepsilon) \succsim{\varepsilon}^{-\tau}L(1/\varepsilon)^{\tau}, \] where, for \(``\Leftarrow"\) the additional assumption \(\varphi(X,\varepsilon)\approx \varphi(X,2\varepsilon)\) is required;
(b) we have \[ e_n(u)\precsim n^{-1/2-1/\tau}L(u)\quad \Leftrightarrow \quad \varphi(X,\varepsilon) \precsim{\varepsilon}^{-\tau}L(1/\varepsilon)^{\tau}. \] In the case of the non-Gaussian symmetric \(\alpha\)-stable vectors the following result is known (see W. V. Li and W. Linde [J. Theor. Probab. 17, No. 1, 261–284 (2004; Zbl 1057.60047)]; F. Aurzada [Probab. Math. Stat. 27, No. 2, 261–274 (2007; Zbl 1136.60033)])
Proposition 2. Let \(X\) be an \(E'\)-valued symmetric \(\alpha\)-stable vector generated by the operator \(u:E\to L_{\alpha}(S,\sigma)\). Let \(\tau>0\) and let \(\theta\in\mathbb R\) be given, where, in addition, \(\tau<\alpha/(1-\alpha)\) for \(0<\alpha<1\). Then:
\[ (a)\quad e_n(u)\succsim n^{1/\alpha-1/\tau-1}(\log{n})^{\theta/\tau} \quad \Rightarrow \quad \varphi(X,\varepsilon) \succsim{\varepsilon}^{-\tau} (-\log{\varepsilon})^{\theta}; \]
\[ (b)\quad \varphi(X,\varepsilon) \precsim{\varepsilon}^{-\tau} (-\log{\varepsilon})^{\theta} \quad \Rightarrow \quad e_n(u)\precsim n^{1/\alpha-1/\tau-1}(\log{n})^{\theta/\tau}; \] the respective converse in the above implications does not hold in general.
This result shows that, unfortunately, only two of the four implications from Proposition 1 can be transferred to the non-Gaussian case. In particular, probably the most interesting and useful implication (upper estimates for \(e_n (u)\) yield those for \(\varphi(X,\varepsilon)\)) is not valid in general. The basic goal of this article is to investigate this implication more thoroughly. The main result of this paper is the following.
Theorem 1.3. Let a symmetric \(\alpha\)-stable \(E'\)-valued vector \(X\) be generated by an operator \(u:E\to L_{\alpha}(S,\sigma)\), where \(\sigma(S)<\infty\). Suppose that \(u\) maps \(E\) even into \(L_{\infty}(S,\sigma)\) and that \[ e_n(u_{\infty})=e_n(u:E\to L_{\infty}) \precsim n^{1/\alpha-1/\tau-1}L(n) \] for some \(\tau>0\) and some slowly varying function \(L\) such that \(L(t)\approx L(t^p)\) for all \(p>0\). Then, \[ \varphi(X,\varepsilon) \precsim{\varepsilon}^{-\tau} L(1/{\varepsilon})^{\tau}. \] This theorem gives new estimates for the small deviation rate for several examples of symmetric \(\alpha\)-stable processes, including unbounded Riemann-Liouville processes, weighted Riemann-Liouville processes and the (\(d\)-dimensional) \(\alpha\)-stable sheet.