Lower bounds for the variance of sums of weakly dependent variables
Let \(H\) be the Hilbert space of random variables defined on a common probability space, with zero mean and finite second moment. Consider a sequence \((\xi_n)_{n \geq 1}\) of elements of \(H\). Denote by \(H(A)\) the subspace of \(H\) spanned by the elements \(\xi_i\), \(i \in A\). Let \((\rho (n))_{n \geq 1}\) be a nonincreasing sequence such that \[ 0 \leq \rho (n) \leq \sup_{k \geq 1} \sup \biggl \{E \xi \eta/(E \xi^2E \eta^2)^{1/2} : \xi \in H \bigl( [1, \ldots, k] \bigr),\;\eta \in H \bigl( [k + n, \ldots] \bigr) \biggr\} \] for all \(n \geq 1\). It is shown that if \(\rho (1) < 1\) and \(\rho (n) \to 0\) as \(n \to \infty\), then there exist a constant \(C_1 = C_1((\rho (n))_{n \geq 1}) > 0\) and an absolute constant \(C_2 > 0\) such that \[ E \left( \sum^n_{k = 1} \xi_k \right)^2 \geq C_1 \exp \left\{ - C_2 \sum^n_{k = 1} k^{-1} \rho (k) \right\} \sum^n_{k = 1} E \xi^2_k. \]
scientific article; zbMATH DE number 794629