Well, first of all, triangular arb isn't what you call a "statistical arb". Triangular arb is "pure arbitrage", meaning that free money exists in a given time frame, assuming you can fill all three trades at once.
Statistical arbitrage, as the name indicates, is based on some kind of statistics. The word "arbitrage" get's thrown around a lot and it doesn't necessarily mean anything out of context.
Simplest thing is to assume that you have a co-integration between two instruments, say A and B. You believe A is relatively overvalued and/or B is relatively undervalued. Therefore you short A and/or go long in B, expecting them to converge over time. Depending on your strategy, parameters or instruments, that may take seconds, hours, days or weeks to happen.
A/B can be literally anything, so just from the top of my head:
- Silver futures vs silver miners. One can expect that the price of silver and silver miners earnings (and therefore stock price) are correlated.
- Old crop / new crop futures spread in wheat. You can look at the historical price difference between last futures before the crop and first month after the crop has arrived.
- LIFFE vs MATIF wheat futures have different specifications. You can look at the historical relation and if the price difference diverges to some significance, you'll take a position expecting it to revert.
- "Upstream/Downstream" and Crack spreads. Oil refineries don't care about price of oil and distillates as much as they care about their profits. So you'd expect them to hedge their "crack spread" and you can take positions, expecting it to revert to some certain level when it's too out of whack. Read more here - Crack spread