  It is well known that Hoeffding's inequality has a lot of applications in the
signal and information processing fields. How to improve Hoeffding's inequality
and find the refinements of its applications have always attracted much
attentions. An improvement of Hoeffding inequality was recently given by Hertz
\cite{r1}. Eventhough such an improvement is not so big, it still can be used
to update many known results with original Hoeffding's inequality, especially
for Hoeffding-Azuma inequality for martingales. However, the results in
original Hoeffding's inequality and its refinement one by Hertz only considered
the first order moment of random variables. In this paper, we present a new
type of Hoeffding's inequalities, where the high order moments of random
variables are taken into account. It can get some considerable improvements in
the tail bounds evaluation compared with the known results. It is expected that
the developed new type Hoeffding's inequalities could get more interesting
applications in some related fields that use Hoeffding's results.
