A Practical Lesson in Bounded Random Walks

One of the tenets of orthodox financial theory is that stock prices in liquid markets move according to a random walk, and therefore the market’s future cannot be profitably predicted.

The notion of a random walk itself is sometimes better known as the “drunkard’s walk”, which is the idea that a drunk who starts at a lamppost and staggers, one step at a time, each step in a random direction, will pass the lamppost an infinite number of times, albeit each time longer apart.

An interesting (and more market-relevant) variant, popularized by paleontologist Stephen Jay Gould in an evolutionary context, is that of a drunk who starts at a bar that is 30 feet from a gutter. Each time he moves he staggers five feet, sometimes toward the bar, sometimes away. But every time he hits the bar he has to stop, of course, because he can’t go through it.

In the long run, what happens to this drunkard? Eventually, and with 100% certainty, he ends up in the gutter. Despite the motion being random, the mere existence of a boundary on one side causes him to move haltingly gutter-ward, eventually falling in.

Many things that exhibit superficial random walk characteristics are actually more like the drunkard ricocheting off the bar. That is, they are bounded, which is crucial to understanding some system dynamics.

I got to thinking about all of this about 4am last night while sharing a bed with my five-year old. I was on the left side, he was on the right, and there was a wall beside him. About four times during the night he ended up on top of me, which irritated me to no end, especially at 4am. And then, of course, it struck me: This was a classic right-bounded random walk. While he was going back-and-forth randomly, he could still no more avoid rolling over me than Gould’s drunkard can avoid falling in the gutter.

Scant reassurance, for sure, in the wee hours, but you have to find some solace when being smeared by a rolling five-year-old.


  1. I call bologne on your theory. I can attest to the fact that my daughter with or without a wall on her side will end up on top of me. And after working through the math I can also definitively state that the probability that her foot will end up in my face is much higher than can be accounted for by an random walk theory.

  2. fewquid says:

    2 questions:
    1) Do the odds of toe-in-nose decrease as the child gets older? At 21 months it is a core compentency for my daughter.
    2) Why last night? Seems the little beasties were synchronized. It’s a conspiracy!

  3. Hi Paul,
    Thanks for the theory and the story. (smile) Yours and the reader “dad”‘s comment got me thinking that you probably don’t have to have a physical wall. With the added “risk” and pain from falling off a bed, the probability of “foot in face” will need to be adjusted with the subconscious risk factor in the daughters/sons’ mind. (smile)
    – Kempton

  4. As a parent of 9 and 12 year olds, I have many years of study on the topic. Known as “Curly’s Law,” the motion of a sleeping child is actually circular, akin to the rotation of a pinwheel, with the child on their side traveling in a clockwise direction (in the northern hemisphere) about once per hour. Driven by impulses deep in the cortex, it’s an evolved practice and survival instinct to hog the comforter and any other form of warmth from all others. Hence it’s entirely likely to get toe in nose, knee in groin, elbow to nose bridge, and head butt several times in the course of a night’s rest(?).

  5. Thanks Drake. You just caused me to laugh out loud and spit water on a table during a meeting.

  6. I have to say, Paul, your financial theory classes are *way* more interesting — and easier to understand — than the ones I remember from school :-)