Abstract This is a short survey about asymptotic properties of a supercritical branching process (Z_{n}) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale (W_{n}) , about...

We present limit theorems for locally stationary processes that have a one sided time-varying moving average representation. In particular, we prove a central limit theorem (CLT), a weak and a strong law of large numbers (WLLN, SLLN) and a law of the iterated logarithm (LIL) under mild...

In a recent paper by Chen (in press) the limit law of the iterated logarithm for the partial sums of i.i.d. real-valued random variables has been established. In this note we look at the corresponding problem in Banach space setting. Let (B,‖⋅‖) be a real separable Banach space with...

We study sharp square function inequalities for martingales taking values in a 2-convex Banach space B. We show that an appropriate weak-type bound holds true if and only if B is isometric to a Hilbert space.