continuously tough [screenshots]

As I was looking to a pointer on continuous random variables for an X validated question, I came across those two “definitions”…

borderline infinite variance in importance sampling

As I was still musing about the posts of last week around infinite variance importance sampling and its potential corrections, I wondered at whether or not there was a fundamental difference between “just” having a [finite] variance and “just” having none. In conjunction with Aki’s post. To get a better feeling, I ran a quick experiment with Exp(1) as the target and Exp(a) as the importance distribution. When estimating E[X]=1, the above graph opposes a=1.95 to a=2.05 (variance versus no variance, bright yellow versus wheat), a=2.95 to a=3.05 (third moment versus none, bright yellow versus wheat), and a=3.95 to a=4.05 (fourth moment versus none, bright yellow versus wheat). The graph below is the same for the estimation of E[exp(X/2)]=2, which has an integrand that is not square integrable under the target. Hence seems to require higher moments for the importance weight. Hard to derive universal theories from those two graphs, however they show that protection against sudden drifts in the estimation sequence. As an aside [not really!], apart from our rather confidential Confidence bands for Brownian motion and applications to Monte Carlo simulation with Wilfrid Kendall and Jean-Michel Marin, I do not know of many studies that consider the sequence of averages time-wise rather than across realisations at a given time and still think this is a more relevant perspective for simulation purposes.