  This paper extends and complements the existing theory for the parabolic Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of $p$-Laplace type. The main results include characterizations for the limiting parabolic $A_\infty$ and $A_1$ classes by applying an uncentered parabolic maximal function with a time lag. Several parabolic Calder\'on-Zygmund decompositions, covering and chaining arguments appear in the proofs. 
Characterizations of parabolic Muckenhoupt classes