Random walks sit underneath a large fraction of quant interview questions, including many that do not look like random walks.
The basics
A symmetric walk takes steps of plus or minus 1 with equal probability. After n steps:
- E[position] = 0, by symmetry.
- Var(position) = n, since steps are independent with variance 1 each.
- Typical distance from the origin is therefore of order sqrt(n).
That last point is the intuition to carry: distance grows like the square root of time, not like time. After 100 steps you are typically about 10 away, not 50.
Expected absolute distance
A common follow-up. E[|position|] after n steps is approximately sqrt(2n/pi), which is roughly 0.8 sqrt(n). Knowing the sqrt(n) scaling and that the constant is a bit under 1 is usually enough.
Recurrence
A classic result worth knowing verbatim, because interviewers ask it directly:
- One dimension: returns to origin with probability 1.
- Two dimensions: returns with probability 1.
- Three dimensions and above: does not - roughly 34% return probability in three dimensions.
The memorable summary, attributed to Kakutani: a drunk man will find his way home, but a drunk bird may not.
Return times
The subtle one. In one dimension the walk returns to the origin with probability 1, but the expected time to return is infinite.
That combination surprises people and is a genuinely good test of whether someone understands the difference between "happens almost surely" and "happens in reasonable time". The return-time distribution has such a heavy tail that its mean diverges.
The reflection principle
For questions of the form "what is the probability the walk stayed positive throughout", the reflection principle maps bad paths onto a set you can count. It underlies the ballot problem and the arcsine laws.
The idea: reflect the portion of a path after its first touch of a level, which pairs each violating path with exactly one path ending elsewhere, turning a constrained count into an unconstrained one.
Why it matters beyond the interview
Price paths are modelled this way, gambler's ruin is a walk between barriers, and inventory on a market-making desk is a walk you are paid to control. The interview questions are not arbitrary.
Practise in stochastic processes.