A probabilistic inequality. (English. Russian original) Zbl 0738.60010

Braverman, M. Sh.

Theory Probab. Math. Stat. 43, 29-34 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 26-32 (1990).

Summary: Let \(\{X_ k\}^ \infty_{k=1}\) be a sequence of independent random variables with identical symmetric distribution such that \(E| X_ 1|^ q<\infty\), where \(0<q\leq 2\). It is shown that under certain restrictions on the distribution of these variables the inequality \[ P\left[\left|\sum^ n_{k=1}a_ kX_ k\right|\left(\sum^ n_{k=1}| a_ k|^ q\right)^{-1/q}\geq x\right]\geq BP[| X_ 1|\geq x] \] holds for \(x\geq 1\), where the constant \(B\) is independent of \(n\) and the numbers \(a_ k\).

MSC:

60E15	Inequalities; stochastic orderings
60G60	Random fields

Keywords:

Bernstein inequality