Some local results on the convergence and the quadratic variation of random sequences (Q1824275)

We study the following property \({\mathcal P}\) of a random sequence \((X_ n\), \(n\in {\mathbb{N}}):\) \(X_ n\) converges a.s. to a finite limit and its quadratic variation is finite a.s.. Several local results on the property \({\mathcal P}\) are established for random sequences satisfying \[ E(X_{n+1}| {\mathcal F}_ n)\leq X_ n+h_ n \] where \((h_ n\), \(n\in {\mathbb{N}})\) is a random sequence. As an application random sequences taking values in a closed, possibly unbounded, interval of \({\mathbb{R}}\) are studied.