This is a qualitative question. In the literature of particle filters/ sequential monte carlo, particle degeneracy is unavoidable. However, frequented cases where it may happen is when the likelihood is very peaked. The result of this
Degenracy is usually explained as the all the weights becoming extremely small and it becomes very sensitive due to the peaked likelihood. Therefore, when we normalise we get all the weights accumulated by 1 particle or a few particles from the entire particle set. Then it is state that this idea of degeneracy corresponds to the $var(w^i_t)$ being extremely high.I do not get the logical deduction that degeneracy ->var(w^i_t) becoming high.
Do they mean that the value of $w^i_t$ is extremely sensitive, meaning a slight nudge to particle $x^i_t$ in a particular direction(closer to the likelihood peak) will result in the $w^i_t$ having high weights all of a sudden. Or could it mean that if I repeated an experiment for another time that $w^i_t$ is likely to have a different value?
Can someone give a mathematical formulation of this?