The Lyapunov theorem for measures valued in Orlicz sequence spaces (Q5928184)

\textit{V. M. Kadets} and \textit{G. Schekhtman} [St Petersburg Math. J. 4, No. 5, 961-966 (1993; Zbl 0804.46048)] generalized the classical Lyapounov convexity theorem by showing that the closure of the range of every nonatomic measure with values in \(X=\ell_p\), \(1\leq p<+\infty\), \(p\not=2\), or in \(X=c_0\) is convex. A Banach space \(X\) with this property is said to be a Lyapunov space. In the paper under review, the author shows that, if \(M\) is an Orlicz function satisfying the \(\Delta_2\)-condition and \(2\notin [\alpha_M,\beta_M]\), then \(X=\ell_M\) is a Lyapunov space, and more generally \(X=(X_1\oplus X_2\oplus\cdots)_{\ell_M}\) is a Lyapunov space if \(X_1, X_2,\ldots\) are Lyapunov spaces. The proof uses, among others, the following lemma, from the quoted paper of Kadets and Schechtman: Let \((r_n)_{n\geq 1}\) be the sequence of the Rademacher functions; consider the stopping time \(T(\omega)=\inf\{j\geq 1\); \(|\sum_{i=1}^j r_i(\omega)|\geq \sqrt N\}\), and set \(s_i(\omega)=r_i(\omega)\) if \(i\leq T(\omega)\), and \(s_i(\omega)=0\) if \(i>T(\omega)\). Then: 1) \(\lim_{N\to+\infty} {\mathbf P}(|\sum_{i=1}^k s_i|<\sqrt N)=0\) if \(N=o(k)\); 2) \(\lim_{N\to+\infty} {\mathbf P}(|\sum_{i=1}^k s_i|<\sqrt N)=1\) if \(k=o(N)\).