A random CLT for dependent random variables
The main results in the paper are as follows: Let \(\{Y_ n\}\) be a sequence of random elements in a metric space (S,d) and \(\{T_ n\}\) be a sequence of random indices. Suppose, for a non-decreasing sequence of positive numbers \(\{k_ n\}\), a non-decreasing sequence of positive integers \(\{a_ n\}\) and a non-increasing sequence of positive numbers \(\{\delta_ n\}\), \(P(| k_{T_ n}-k_{a_ n}| >\delta_{a_ n}k_{a_ n})\to 0\) as \(n\to \infty\). Then \(Y_{T_ n}\) converges weakly to an S-valued random element Y \((Y_{T_ n}\Rightarrow Y)\) if \(Y_{a_ n}\Rightarrow Y\) and \[ (C)\quad \max_{i}\{d(Y_ i,Y_{a_ n}):| k_ i-k_{a_ n}| \leq \delta_{a_ n}k_{a_ n}\}\to 0\quad as\quad n\to \infty. \] Condition (C) is also necessary when S is separable. Further, (C) is both necessary and sufficient for \(d(Y_{T_ n},Y_{a_ n})\to^{p}0.\) These results have been applied to the CLT for a general class of sequences of dependent random variables.
scientific article; zbMATH DE number 4009426