Towards the question of $CPT$-invariant theories of infinite-component fields

A study is made of the problem of the $CPT$-invariaace of the theory of infinite-component fields by a method close to the one developed by Pauli. We consider only Lagrangians constructed from bilinear tensor forms, Such Lagraugians are $CPT$-invariant if for fields which transform in accordance with representations of the proper Lorentz group from the classes (A), (C), (D) (in the classification of Gel'land and Yaglom) we assume the ordinary connection between spin and statistics, and if for fields of the class (B) we assume that the statistics are defined not by the spin but by the numbers $k_1$ ($k_1$ together with the least spin $k_0$ characterizes an irreducible representation of the proper Lorentz group). It is also shown that fields of the class (E) admit $CPT$-noninvariant theories.