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What is Random Walk Theory?

A random walk refers to any process in which there is no observable pattern or trend; that is, where the movements of an object, or the values taken by a certain variable, are completely random. Certain real-life scenarios that could be modeled as random walks could be:

• The movements of an animal foraging for food in the wilderness
• The path traced by a molecule as it moves through a liquid or a gas (diffusion)
• The price of a stock as it moves up and down
• The path of a drunkard wandering through Greenwich Village
• The financial status of a gambler at the roulette wheel in Las Vegas

To illustrate the concept of random walk a bit further, let’s look at the simplest random walk: a random walk along on the integer number line.

A Random Walk on the Number Line

Consider the standard integer number line:

random walk theory

Let’s talk a random walk along the integer number line. Image by Martin Smith-Martinez.

Imagine you are standing on zero. You flip a fair coin, and if it lands on heads, you will take one step (one unit) to the right. However, if the coin lands tails, you will take one step to the left, again, one unit. Therefore, with each coin toss there is a 50% chance you will move to the right, and a 50% chance you move to the left.

Let’s now imagine you flip the coin ten times, and you get the following outcome:

HHHTHHTTTT

This would generate the following random walk: right, right, right, left, right, right, left, left, left, left. We can illustrate it as follows:

random walk theory

Our random walk of 10 steps, illustrated.

In this case, we happen to end up back where we started: at zero. This random walk happened to work out that way, but it need not have dropped us off where we began. However, it turns out that because the probabilities of moving right or left are equal, we have a symmetrical random walk, where we will, on average, end up back where we started, at zero.

For more on the average displacement of a random walk, check out this very readable page on random walks from MIT.

Random walks and root-mean-square distance

For a random walk like the one described above, it turns out that after taking n steps, we will be approximately a distance of √n away from the origin (zero). √n is known as the root-mean-square distance. Try the following for practice:

• For a symmetrical random walk on the integer number line, how far away from zero can you expect to be after 49 steps? After 100 steps? After 256 steps?

• Answer: 7, 10, 16.

By taking the square root of the number of steps, we find about how far away we’ll be, on average, from the initial state. Note that as n gets very large, we expect to get further and further away from the origin.

For the derivation of the root-mean-square speed, check out this page from the University of Virginia. Otherwise, read on for a fascinating conclusion about random walks that involve an infinite number of steps.

A Fun Caveat on Random Walks: Monkeys and Typewriters

From the discussion on root-mean-square distance, we see that as n gets larger, we expect to get further and further from the origin. In fact, for an infinite number of random walks with an infinitely large number of steps, we will end up visiting every number on the number line.

This idea was illustrated quite vividly and humorously in the book Fooled by Randomness by former options trader Nassim Nicholas Taleb. In that book, Taleb posits the idea that if a sufficiently large group of monkeys were to type randomly on typewriters (a more complex variety of a random walk) for an infinite amount of time, one of those monkeys would necessarily compose The Iliad. And not only that—given infinite time, one of the monkeys who composes the Iliad will then go on to compose the Odyssey. We would undoubtedly hail such a monkey as a genius—when in reality it was just randomness, not literary talent, that led the ape to pen the classic tome. In fields that involve high degrees of randomness, like finance, Taleb warns us against thinking too highly of so-called “genius traders” who are making a killing in the markets—like our Homeric monkey, they might just be getting lucky.

random walk theory

Monkeys writing on typewriters constitute a more complex type of random walk. Given an infinite number of them and an infinite amount of time, some of these apes will outdo Shakespeare. Photo by New York Zoological Society.

Random Walk Theory in Finance

Perhaps the best and most widely known application of random walk theory is in finance. Random walk theory was first popularized by the 1973 book A Random Walk Down Wall Street by Burton Malkiel, an economics professor at Princeton University. The crux of the theory is that the price fluctuations of any given stock constitute a random walk, and therefore, future price movements cannot be predicted with any accuracy. (Sorry, stock forecasters!)

To test this theory, Malkiel had his students create a random walk very similarly to the one we created above. Malkiel had the students imagine a hypothetical stock, call it XYZ, and flip coins to determine the performance of the stock. If the coin landed heads, that indicated stock XYZ closed higher on the day. If the coin landed tails, XYZ closed lower. After sufficient coin flips, Malkiel’s students graphed the price performance of XYZ, in a chart much like the following:

random walk theory

A hypothetical stock with price movements generated by coin flipping; a “random walk” of price movements. Image by Althepal.

Malkiel then showed this to a stock technical analyst—a type of stock forecaster who looks at charts of stock price movements, and makes buy and sell predictions based on the patterns of the chart. This particular analyst took one look at Malkiel’s chart, and concluded the stock was an immediate buy. However, the pattern generated was completely random—a random walk!

Random Walk Theory: A conclusion

A random walk is the random motion of an object along some mathematical space. Like much of statistics, random walk theory has useful applications in a variety of real-world fields, from Finance and Economics to Chemistry and Physics. For more on random walks, check out our statistics blog and videos!

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