During Johan Seger’s seminar in Warwick, on the control variate improvements he developed with Rémi Leluc (which PhD thesis committee I joined), Aymeric Dieuleveut, François Portier, and Aigerim Zhuman, I started wondering at whether or not a control variate could turn an infinite variance Monte Carlo estimate into a finite variance one. And asked… ChatGPT about it, with the above reply that is correct if not practical in the least since the example provided therein was reverse-engineering an infinite variance rv into a sum of an infinite variance rv considered as the control variate and a finite variance rv. As summarised below. In practice, this would mean replacing the integrand of interest with a much simpler integrand that shares the same asymptotic behaviour, not an easy task! (As an aside, I found out that enabling MathJax on this ‘Og would cost me $40 a month!)
5. Summary
✅ Theoretical possibility:
Yes — control variates can make an infinite-variance estimator finite, but only if the control’s sample path shares the same tail driver and its expectation is known.infinite variance rv
In real-world Monte Carlo, when X is heavy-tailed, you usually:
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Split X = Y + (X-Y), where Y has known expectation and similar tails,
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Use Y as control variate, and
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Possibly combine with truncation, conditional expectation, or importance sampling for stability.
