When Does Russian Roulette Have a Mean? Exact Moment Boundaries for an Adjacent-Prefix Generative Objective

Training or evaluating a latent-variable model often requires the log probability of an observation, but the integral over the hidden variable is unavailable. Importance sampling replaces that integral by an average of positive weights. Taking the logarithm makes the estimate biased. The Stochastically Unbiased Marginalization Objective (SUMO) tries to remove this bias as follows: compare estimates made with n and n-1 shared samples, divide each difference by the probability of reaching level n, and stop at a random level. The usual argument checks each correction separately. That is not enough: every run can stop and return a finite number while the distribution across runs has no mean. We answer exactly when a mean exists for the original adjacent-prefix construction when the weights are bounded, positive, and nonconstant. We give necessary-and-sufficient conditions for an ordinary mean and for every fractional moment of order $1