Blackwell's renewal theorem for certain linear submartingales and coupling (Q1323524)

The paper is devoted to the extensions of the Blackwell's renewal theorem from the classical case of nonnegative summands \(X_ n\), \(n\geq 1\), to the more general situation. The sequence \((X_ n)\) is assumed to satisfy two main conditions: stochastic boundedness, i.e. \[ G(t)\leq P(X_{n+1}\leq t\mid{\mathcal F}_ n)\leq F(t) \] for some distributions \(F\) and \(G\) with finite means; and stochastic stability, i.e. \[ \lim_{k\to\infty} \sup_{n\geq 0} \biggl\| k^{-1} \sum^ k_{j=1} E(X_{n+j}\mid{\mathcal F}_ n)-\theta \biggr\|_ \infty=0, \] for some \(\theta\in{\mathbb{R}}\), where \(({\mathcal F}_ n)_{n\geq 1}\) is the corresponding filtration and \(\|\cdot\|_ \infty\) denotes the \(L_ \infty\)-norm. Proofs of the results are based on the coupling method. A set of examples is also given.