For a measure preserving automorphism $T$ of a probability space, we provide conditions on the tail function of $g\colonΩ\to\mathbb R$ 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.