Rigorous exponential decay of finite-N bias in the INS Fleming–Viot estimator

Establish that the finite-N bias of the time-averaged empirical measure estimator for ψ^ε(x)φ^ε(y) constructed from N forward–backward Fleming–Viot particle pairs with swapping (as defined in Section 3.3 via the indicator-weighted measure over (X_t^{(n),K}, Y_t^{(n),K}) and S_t^{(n),K}) decays exponentially in N, by providing a complete rigorous proof and quantitative bounds under the assumptions stated for the interacting particle system and its infinite-swapping limit.

Background

In Section 3.3, the authors construct an estimator for ψ^ε(x)φ^ε(y) using N pairs of particles in forward and backward Fleming–Viot systems with swapping. The estimator relies on a time-averaged empirical measure over the paired particle positions, weighted by the current forward/backward assignment variable S_t^{(n),K}.

They note that for finite N this estimator is biased but the bias vanishes as N→∞. A formal large deviations argument suggests that the bias decreases exponentially with N, motivating a rigorous analysis to confirm this decay rate and to provide precise bounds. This would strengthen theoretical guarantees for the interacting particle approximation of the nonlinear Markov process and its INS limit.