  Ever since the proof of asymptotic normality of maximum likelihood estimator
by Cramer (1946), it has been understood that a basic technique of the Taylor
series expansion suffices for asymptotics of $M$-estimators with
smooth/differentiable loss function. Although the Taylor series expansion is a
purely deterministic tool, the realization that the asymptotic normality
results can also be made deterministic (and so finite sample) received far less
attention. With the advent of big data and high-dimensional statistics, the
need for finite sample results has increased. In this paper, we use the
(well-known) Banach fixed point theorem to derive various deterministic
inequalities that lead to the classical results when studied under randomness.
In addition, we provide applications of these deterministic inequalities for
crossvalidation/subsampling, marginal screening and uniform-in-submodel results
that are very useful for post-selection inference and in the study of
post-regularization estimators. Our results apply to many classical estimators,
in particular, generalized linear models, non-linear regression and cox
proportional hazards model. Extensions to non-smooth and constrained problems
are also discussed.
