Mathematics & Statistics

Markov Chains: Only Now Matters

Andrei Markov's 1906 model: a system where the next state depends only on the current state, not the full history. Weather models, Google's PageRank, and language models all use this principle.

Ergodicity: Why Average Outcomes Are Not Your Outcomes

An ergodic system is one where the average across many people equals the average over time for one person. Most real-world risks aren't ergodic — and ignoring this leads to ruin, even when expected value looks fine.

Fat-Tailed Distributions: The Extreme Is Normal

In a normal distribution, extreme events are vanishingly rare. In a fat-tailed distribution, they're not — and most risk models, built on normal assumptions, dangerously underestimate tail risk.