Occupation times of discrete-time fractional Brownian motion

We prove a conditional local limit theorem for discrete-time fractional Brownian motions (dfBm) with Hurst parameter [Formula: see text]. Using results from infinite ergodic theory, it is then shown that the properly scaled occupation time of dfBm converges to a Mittag-Leffler distribution.

Symmetric Birkhoff sums in infinite ergodic theory

We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity. Also, we consider the latter phenomenon and characterize it among transformations admitting generalized recurrent events.