We present a non-convergence theory for probabilistic direct search, a randomized
derivative-free optimization method, where non-convergence means the failure to produce
iterates that achieve stationarity asymptotically. The motivation is to understand whether
the submartingale-like assumption in the existing convergence theory is essential or merely
an artifact of the analysis techniques. For convex objectives, we prove that the probability of
non-convergence is positive, provided that the polling directions satisfy a probabilistic ascent
condition that is essentially the opposite of the submartingale-like convergence condition.
Furthermore, we establish a lower bound for the non-convergence probability. For the typical
implementation of this method, where each iteration draws a fixed number of random polling
directions independently and uniformly from the unit sphere, our theory implies that the
method is not globally convergent if the number of directions is below the threshold specified
in the convergence theory, and the submartingale-like assumption is confirmed to be essential
for convergence. Our theory is obtained by examining two random series that control the
distance from any iterate to the starting point and estimating the probability for these series
to stay below certain bounds.