Local limit theorems and weakly dependent processes (Q2752163)

Let \(X_1,X_2,\dots\) be a stationary symmetric process of real-valued random variables satisfying the strong mixing condition NEWLINE\[NEWLINE\alpha(n)= \sup_{A\in{\mathcal F}(1, k),B\in{\mathcal F}(k+ n,\infty),k\in \mathbb{N}}|P(A\cap B)- P(A) P(B)|\downarrow 0NEWLINE\]NEWLINE or the \(\rho\)-mixing condition, which is more restrictive than the strong mixing condition, NEWLINE\[NEWLINE\rho(n)= \sup_{f\in L_2({\mathcal F}(1,k)), g\in L_2({\mathcal F}(k+ n,\infty)),k\in \mathbb{N}}{Ef(g)\over\|f\|_2\|g\|_2}\downarrow 0,NEWLINE\]NEWLINE where \({\mathcal F}(n,m)\) denotes the \(\sigma\)-algebra generated by \(X_n,\dots, X_m\). Under some additional restrictive assumptions the authors prove a local limit theorem for this process.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00044].