The Arcsine Law: Who Leads a Coin-Flip Race

Flip a fair coin repeatedly, scoring +1 for heads and -1 for tails, and track the running total's sign over a long sequence. Intuition says a fair game should spend roughly half its time with the total positive and half negative — the coin has no memory, so why would one side dominate? The arcsine law says the opposite is typical: the fraction of time spent in the lead is most likely to be close to 0% or close to 100%, and a roughly even 50/50 split is actually one of the least likely outcomes.

Formally, for a symmetric random walk of $n$ steps, the fraction of time the walk spends positive converges to a random variable with the arcsine distribution, whose density $\frac{1}{\pi\sqrt{x(1-x)}}$ is U-shaped — it piles up probability near $x=0$ and $x=1$ and dips to its minimum at $x=0.5$. In plain English: once a fair coin-flip walk gets ahead, it tends to stay ahead for long stretches, because a walk that returns to zero has to fight its own momentum to cross back, so long uninterrupted leads are the norm, not the exception.

The same law explains why streaks in genuinely fair games — a market-neutral book, a coin-flip trading signal with zero real edge — can look like "hot" or "cold" runs lasting far longer than gut instinct expects, purely from randomness with no skill or regime change involved.