  For a measure preserving automorphism $T$ of a probability space, we provide
conditions on the tail function of $g\colon\Omega\to\mathbb R$ and $g-g\circ T$
which guarantee limit theorems among the weak invariance principle,
Marcinkievicz-Zygmund strong law of large numbers and the law of iterated
logarithm to hold for $f:=m+g-g\circ T$, where $(m\circ T^i)\_{i\geqslant 0}$
is a martingale differences sequence.
