If the sequence of coefficients of a random power series has a degenerate tail field, then either its circle of convergence is a natural boundary, or this situation can be achieved by subtracting a fixed series. This generalises the known result for independent coefficient sequences.
P. Holgate. "The Natural Boundary Problem for Random Power Series with Degenerate Tail Fields." Ann. Probab. 11 (3) 814 - 816, August, 1983. https://doi.org/10.1214/aop/1176993530