Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / S / S2 / e / Solution

State-dependent execution times can bias the finite set of completed outputs. A simple counterexample takes $p=q$ uniform on two states and $M=1$, so every proposal is accepted. Suppose processing the first state takes one unit of time and processing the second takes two. Stop at a deadline between one and two units, discarding any unfinished computation. The only possible completed output is the first state; conditional on obtaining a sample, its law is concentrated there rather than uniform.

This demonstrates that a wall-clock cutoff can select outputs according to their processing cost, unlike the fixed proposal budget in (d). It does not imply a permanent asymptotic bias of averages over an indefinitely growing completed sequence: the finite-horizon stopping and censoring are the issue. Without assumptions relating runtime to states, exact finite-sample unbiasedness cannot be guaranteed.