Zeta(2) Variables Covariance

This quant interview question is about understanding dependence, covariance, and expectations in the context of heavy-tailed discrete distributions. It places you in a pure probability setting, away from pricing formulas, to test whether your quant prep includes comfort with zeta distributions and power-law tails that often appear in risk modeling and extreme events.

It trains your grasp of when expectations and higher moments exist, and what that implies for quantities like covariance. You must recognize how tail behavior affects integrability and distinguish formally independent variables from situations where standard identities cannot be used safely.

This matters for quant interviews because real-world portfolios and trading PnLs can have heavy tails. Interviewers want candidates who can spot when standard formulas silently assume finite moments, and who can reason rigorously about edge cases in probabilistic modeling.

What it tests

Covariance quantifies the linear relationship between two random variables, and for independent variables, it is theoretically zero because $\mathbb{E}[AB] = \mathbb{E}[A]\mathbb{E}[B]$. However, this formula assumes that both expectations exist and are finite. In heavy-tailed distributions, such as those with power-law decay, the mean or higher moments may diverge, making the covariance undefined or indeterminate. The underlying structure is that the existence of moments is a prerequisite for many standard probabilistic formulas to be valid. When moments diverge, the usual algebraic manipulations can break down, and one must check moment existence before applying such results.