The fixed-horizon reparameterized variational-training procedure terminates

At $t=0$, the initial parameter is the supplied finite-dimensional real vector in the admissible domain by [F1] and the algorithm's hypotheses. No payoff or gradient has yet been recorded. Thus the procedure has a well-defined finite state before its first round.

Suppose $0\le t<T$ and the state before round $t$ has been obtained with finite parameters and records for all earlier rounds. The next supplied draw has an index in the finite data set. If ${ϵ}_t\in \mathcal{N}$, [F3] defines both records to be zero, so they are finite and no undefined density or derivative is evaluated. If ${ϵ}_t\mathrm{\notin }\mathcal{N}$, [F2] makes the two density values strictly positive and finite, so $h_t$ is a finite real number; the sample-path $C^1$ property gives a finite gradient vector $G_t$ at the current admissible parameter. This constructs finite records in both branches.

The scalar ${\alpha }_t$ and all coordinates of ${\eta }_t,G_t$ are finite, so the finite-dimensional vector sum ${\eta }_t+{\alpha }_t{G}_t$ has finite coordinates. The stated update-domain requirement [F3] puts this parameter ${\eta }_{t+1}$ in the same admissible domain where the next round's hypotheses apply. Each branch in step 1.2 and this update uses only finitely many of the procedure's prescribed operations, with no recursive loop within a round. Thus a completed round produces the next finite admissible state.

By [F5], steps 1.1–2.1 give a completed state after each $t\le T$. After round $T-1$ the prescribed loop ends; there is no further stopping test or unbounded search. The finite records comprise exactly $({\eta }_0,\dots ,{\eta }_T)$ and $(h_t,G_t{)}_{0\le t<T}$, which [F1] instructs the procedure to return. Hence it terminates after exactly $T$ rounds with every stated finite output, for every admissible draw sequence including the null-set branches.