Particle Deprivation and the Cost of Resampling

In a particle filter (Monte Carlo localization), resampling is introduced to combat weight degeneracy — the tendency for all importance weight to collect on a few particles. Yet resampling creates its own failure mode, often called particle deprivation (or sample impoverishment).

Explain this trade-off in depth. Your answer should address: (a) what degeneracy is and why, without resampling, the variance of the importance weights provably grows over time; (b) the mechanism by which resampling causes particle deprivation, and concretely how this can make a localization filter diverge or get 'stuck' (e.g. around the kidnapped-robot problem or with a low-noise / highly peaked measurement model); and (c) at least two distinct, principled mitigation strategies, explaining *why* each one helps.