We say that a random integer variable X is monotone if the modulus of the characteristic function of X is decreasing on . 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 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.
A. Aamand is funded by DFF-International Postdoc Grant 0164-00022B from the Independent Research Fund Denmark. N. Alon is supported in part by NSF grant DMS-1855464, BSF grant 2018267 and the Simons Foundation. J. Houen and M. Thorup are supported by VILLUM Foundatioon grant 16582 (BARC) which also supported A. Aamand in the beginning of this research.
On 22 December 2022 the authors corrected the acknowledgement of the BARC grant to read it more clear.
Anders Aamand. Noga Alon. Jakob Bæk Tejs Houen. Mikkel Thorup. "On sums of monotone random integer variables." Electron. Commun. Probab. 27 1 - 8, 2022. https://doi.org/10.1214/22-ECP500