  Widefield surveys of the sky probe many clustered scalar fields -- such as
galaxy counts, lensing potential, gas pressure, etc. -- that are sensitive to
different cosmological and astrophysical processes. Our ability to constrain
such processes from these fields depends crucially on the statistics chosen to
summarize the field. In this work, we explore the cumulative distribution
function (CDF) at multiple scales as a summary of the galaxy lensing
convergence field. Using a suite of N-body lightcone simulations, we show the
CDFs' constraining power is modestly better than that of the 2nd and 3rd
moments of the field, as they approximately capture the information from all
moments of the field in a concise data vector. We then study the practical
aspects of applying the CDFs to observational data, using the first three years
of the Dark Energy Survey (DES Y3) data as an example, and compute the impact
of different systematics on the CDFs. The contributions from the point spread
function are 2-3 orders of magnitude below the cosmological signal, while those
from reduced shear approximation contribute $\lesssim 1\%$ to the signal.
Source clustering effects and baryon imprints contribute $1-10\%$. Enforcing
scale cuts to limit systematics-driven biases in parameter constraints degrades
these constraints a noticeable amount, and this degradation is similar for the
CDFs and the moments. We also detect correlations between the observed
convergence field and the shape noise field at $13\sigma$. We find that the
non-Gaussian correlations in the noise field must be modeled accurately to use
the CDFs, or other statistics sensitive to all moments, as a rigorous cosmology
tool.
