Limit theorems for methods of summation of independent random variables. I
Let \(Z(t)=\sum^{\infty}_{i=1}c_ i(t)X_ i\), \(t\in T\), be a class of series formed by independent random variables \(X_ i\), with values in a linear measurable space, and real numbers \(c_ i(t)\) such that: 1. for all \(i\geq 1\), \(\lim_{t\to t_ 0}c_ i(t)=0\); 2. for some \(k_ 0\geq 1\) \[ \limsup_{t\to t_ 0}\sum_{k\geq k_ 0}var\{c_ i(t)\}_{i\in I_ k}\equiv C(Z)<\infty, \] where \[ I_ k=(n_{k- 1},n_ k],\quad b_ k=| \inf \{| c_ i(t_ k)|: t\in I_ k\}|^{-1},\quad and \] \[ var\{c_ i(t)\}_{i\in I_ k}=\sum^{n_ k-1}_{n=n_{k-1}+1}| c_ i(t)-c_{i+1}(t)| +| c_{n_ k}(t)| \text{ for some } (n_ k)\subset N\quad and\quad (t_ k)\subset T. \] The methods of summation of sequences of Banach-valued random elements are considered. The equivalence of analogues of the law of large numbers for a family \(\{Z(t), t\in T\}\) is proved. One of the results gives conditions under which from the estimate \(\limsup_{t\to t_ 0}\| Z(t)\| \leq M\) follows that \[ \sum_{n\in N}P(\| \sum_{i\in I_ k}(X_ i-X'_ i)\| \geq \alpha b_ k)<\infty \text{ for } \alpha >4M, \] \(X'_ i\) a random element independent of \(X_ i\) and having the same distribution. Moreover, if the series \[ \sum_{k\in N}P(\| \sum_{i\in I_ k}X_ i\| >\alpha b_ k)<\infty \text{ for each }\alpha >\Lambda, \] and \[ \limsup_{k\to \infty}\max_{i\in I_ k}P(\| S_{n_ k}- S_ i\| >Lb_ k)\leq 1,\quad then: \limsup_{t\to t_ 0}\| Z(t)\| \leq (\Lambda +L)C(Z)\quad a.s. \]
methods of summation of sequences of Banach-valued random elements
scientific article; zbMATH DE number 4038883
scientific article