  We say that a random integer variable $X$ is monotone if the modulus of the
characteristic function of $X$ is decreasing on $[0,\pi]$. This is the case for
many commonly encountered variables, e.g., Bernoulli, Poisson and geometric
random variables. In this note, we provide estimates for the probability that
the sum of independent monotone integer variables attains precisely a specific
value. We do not assume that the variables are identically distributed. Our
estimates are sharp when the specific value is close to the mean, but they are
not useful further out in the tail. By combining with the trick of
\emph{exponential tilting}, we obtain sharp estimates for the point
probabilities in the tail under a slightly stronger assumption on the random
integer variables which we call strong monotonicity.
