  Consider a random walk $S_n=\sum_{i=0}^n X_i$ with negative drift. This paper
deals with upper bounds for the maximum $M=\max_{n\ge 1}S_n$ of this random
walk in different settings of power moment existences. As it is usual for
deriving upper bounds, we truncate summands. Therefore we use an approach of
splitting the time axis by stopping times into intervals of random but finite
length and then choose a level of truncation on each interval. Hereby we can
reduce the problem of finding upper bounds for $M$ to the problem of finding
upper bounds for $M_\tau=\max_{n\le \tau}S_n$. In addition we test our
inequalities in the heavy traffic regime in the case of regularly varying
tails.
