Rate of convergence in the weak law of large numbers in a Banach space
Let \((B,\| \cdot \|)\) be a real separable Banach space, \(L_ 0(B)=L_ 0(\Omega,{\mathcal F},P;B)\) be the set of all B-valued random variables, defined on the probability space (\(\Omega\),\({\mathcal F},P)\), \(X_ 1,...,X_ n\) be independent copies of the B-valued random variable X, \(S_ n(X)=X_ 1+...+X_ n\); EX denote the Bochner mean. Let the numbers p,q be real, the space ''of weak type q'' is \[ \Lambda^ q_ 0(B)=\{X\in L_ 0(B),\quad EX=0\quad and\quad \lim_{t\to 0}t^ qP(\| X\| >t)=0\}, \] and \[ RLLN_{p,q}(B)=\{X\in L_ 0(B),\quad for\quad each\quad \epsilon >0,\quad P(\| S_ n(X)\| >\epsilon n^{1/p})=o(n^{1-q})\}. \] In this paper the author defines type (p,r,q) as follows: Let \(1\leq p<2\), \(p\leq q\leq r\leq \infty\). A Banach space is called a space of type (p,r,q), if there exists a constant \(c>0\) such that for any finite collection \((x_ i)^ n_{i=1}\subset B\), \[ E\| \sum^{n}_{1}r_ ix_ i\| \leq cn^{1/p-1/q}\|| (x_ i)\||_{q,r}, \] where \(\|| (x_ i)\||_{q,r}\) denotes the Lorentz norm. Denote by \({\mathcal D}(p,r,q)\) the class of Banach spaces of type (p,r,q). The main theorem is that: if \(1\leq p<2\), \(1<q<\infty\), then \(\Lambda_ 0^{pq}(B)\subset RLLN(B)\) iff the Banach space B has type (p,\(\infty,pq)\).
scientific article; zbMATH DE number 3965070