  We prove deviation bounds for the random variable $\sum_{i=1}^{n} f_i(Y_i)$
in which $\{Y_i\}_{i=1}^{\infty}$ is a Markov chain with stationary
distribution and state space $[N]$, and $f_i: [N] \rightarrow [-a_i, a_i]$. Our
bound improves upon previously known bounds in that the dependence is on
$\sqrt{a_1^2+\cdots+a_n^2}$ rather than $\max_{i}\{a_i\}\sqrt{n}.$ We also
prove deviation bounds for certain types of sums of vector--valued random
variables obtained from a Markov chain in a similar manner. One application
includes bounding the expected value of the Schatten $\infty$-norm of a random
matrix whose entries are obtained from a Markov chain.
