This paper provides an overview of results pertaining to moment convergence for certain ratios of random variables involving sums, order statistics and extreme terms in the sense of modulus. Most of the literature on this matter originates from Darling (1952) who gave a criterion for the convergence in probability to 1 of the ratio of the maximum to the sum in case of nonnegative random variables.
Ladoucette, S. A., & Teugels, J. L. (2007). Asymptotics for ratios with applications to reinsurance. Methodology and Computing in Applied Probability, 9(2), 225-242. https://doi.org/10.1007/s11009-007-9020-z