We study the asymptotic behavior of stream cipher security mea-
sures associated with classes of sequence generators such as linear feedback shift registers and feedback with carry shift registers. For nonperiodic sequences we consider normalized measures and study the set of accumulation points for a fixed sequence. We see that the set of accumulation points is always a closed subinterval of $[0, 1]$. For binary or ternary FCSRs we see that this interval is of the form $[B, 1-B]$, a result that is an analog of an earlier result by Dai, Jiang, Imamura, and Gong for LFSRs.