  Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent but not necessarily
identically distributed random variables. In this paper, the sufficient
conditions are found under which the tail probability
$\mathbb{P}(\sup_{n\geqslant0}\sum_{i=1}^n\xi_i>x)$ can be bounded above by
$\varrho_1\exp\{-\varrho_2x\}$ with some positive constants $\varrho_1$ and
$\varrho_2$. A way to calculate these two constants is presented. The
application of the derived bound is discussed and a Lundberg-type inequality is
obtained for the ultimate ruin probability in the inhomogeneous renewal risk
model satisfying the net profit condition on average.
