  Cohen and Kontorovich (COLT 2023) initiated the study of what we call here
the Binomial Empirical Process: the maximal absolute value of a sequence of
inhomogeneous normalized and centered binomials. They almost fully analyzed the
case where the binomials are independent, and the remaining gap was closed by
Blanchard and Vor\'{a}\v{c}ek (ALT 2024). In this work, we study the much more
general and challenging case with correlations. In contradistinction to
Gaussian processes, whose behavior is characterized by the covariance
structure, we discover that, at least somewhat surprisingly, for binomial
processes covariance does not even characterize convergence. Although a full
characterization remains out of reach, we take the first steps with nontrivial
upper and lower bounds in terms of covering numbers.
