Teoremi limiti funzionali per sistemi moltiplicativi. Nota I

Gonzalez Villalobos, Alvaro

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 55 (1973) no. 3-4, pp. 228-235

Résumé

In this paper we prove some theorems on weak convergence in $(D,\mathcal{D})$, or $(C,\mathcal{C})$, of random elements $X_{m}$ to a Gaussian process. $D$ denotes the set of real-valued functions defined on $[0,1]$ that are right-continuous and have left-hand limits, $\mathcal{D}$ the Skorohod $\sigma$-field in $D$ (cfr. [3], p. 111), $C$ the set of real-valued continuous functions on $[0,1]$, and $\mathcal{C}$ the $\sigma$-field that gives the uniform topology in $C$; moreover, $X_{m} \Rightarrow X$ will denote weak convergence. The random elements $X_{m}$ are constructed on the basis of the linear means of the series $S(w) = \sum_{j=1}^{\infty} a_{j} \psi_{j}(w)$ where $\{ \psi_{j}\}$ is a uniformly bounded equinormed strongly multiplicative system. These theorems generalize analogous results (cfr. [4]) valid for the case of a lacunary trigonometric series $S(w) = \sum_{j=1}^{\infty} a_{j} \cos n_{j} w$ with lacunarity $q \ge 3$.