Breaking the Bell Curve

Natural Sciences9 min. Read

Professor and student push limits of foundational math theorem

Evan Randles, assistant professor of mathematics
Evan Randles, assistant professor of mathematics.
By Christina Nunez
September 2, 2021

Flip a coin. Then flip it again. And again. How many times would that coin land on heads or tails over the course of, say, a thousand tosses? The central limit theorem, a bedrock result in statistics and probability theory, reliably describes the answer.

But there are many “what ifs”—conditions that go outside the bounds of a basic approach to the central limit theorem.

Evan Randles, assistant professor of mathematics, has been working on those “what if” problems for the last several years. Recently, he and Huan Bui ’21 achieved a breakthrough that became the basis of Bui’s honors thesis. The two also authored a paper that is currently under review for publication in a leading mathematical journal.

Their paper articulates a scenario, illustrated with animated visualizations, of how the theorem plays out when probabilities take on complex values in expanded dimensions. The mathematics involved has relevance for wide-ranging practical applications such as data analysis and climate modeling.

“Working with Huan has been a highlight of my career,” Randles said. “It was the most rewarding experience to have a student who was so eager and so excited and worked so hard.”

Evan Randles, assistant professor of mathematics
Evan Randles, assistant professor of mathematics, and Huan Bui ’21 achieved a breakthrough in the central limit theorem that visualizes how the theorem plays out when probabilities take on complex values in expanded dimensions.

If you were to graph the results of the central limit theorem, which has been around since the turn of the 19th century, you would get a bell curve. In the coin-toss example, the center of the bell tells you it’s highly probable you’ll get some relatively even mix of heads and tails. The tapering sides of the bell capture the less-likely scenarios, such as landing on heads 900 out of 1,000 times.

Mathematically, you could define this game with a function that serves as the “rules.” One function could define how much money you’ll win every time the coin lands on heads, for example. Another function could describe an entirely different game, perhaps, using a trick coin or rolling a die.

“What’s amazing about the central limit theorem is that it doesn’t really matter what the rules of the game are,” Randles said. “You’re always going to see a bell curve.”

Bell curve graphic
Figure 1: Consider a random walker who is equally likely to move up, down, left, or right. The probability distribution of their location after a fixed number of steps resembles a physical bell. This example is a two-dimensional analog of successively tossing a fair coin. The distribution evolves, via the convolution, as the walker takes more steps.

Continuing to successively play the game is known, mathematically, as convolution—a basic ingredient of the central limit theorem. The question mathematicians have wrestled with in some form or another for more than 200 years is, just how far can you push the central limit theorem before that dependable bell curve starts to change shape?

Randles is not the first Colby mathematician to work on this question. I.J. Schoenberg, who taught at Colby in the 1930s, was among those to prove that indeed, it was possible to get more than just a bell curve with a succession of many convolutions. Randles cited Schoenberg in a 2015 paper he coauthored with his Ph.D. advisor at Cornell University, Laurent Saloff-Coste. In that paper, he and Saloff-Coste described various outcomes produced by introducing negative numbers to this mathematical landscape.

The work with Bui began with conversations during office hours. “I would ask him to read something, and then he would come back and chat in my office for like four hours about it,” Randles said. After a few months, they began talking about working on a problem together.

Bui majored in physics, but he ended up getting “slowly sucked into the Math Department,” he said, and added math as a second major. He is now at the Massachusetts Institute of Technology working on his Ph.D. in experimental atomic physics.

Randles had extended his work with Saloff-Coste in another paper. While the original bell curve distribution is one-dimensional, the corresponding distribution predicted in two dimensions looks like a physical bell. Randles began to ramp up the complexity, exploring the reaches of the central limit theorem with complex probabilities in higher dimensions.

Going back to the coin toss, this is a bit like flipping multiple coins in successive games and calculating a result. Under those conditions, Randles asked, what type of “bells” can be produced?

Bell curve graphic
Figure 2: When “probabilities” are complex numbers, the familiar bell curve is replaced by more curious shapes.

The graphical side of the problem drew Bui in. He began trying to reproduce graphs he saw in Randles’s previous papers. “What we got was some funky-looking graphs that are very interesting,” Bui said. “That got me interested in the problem itself—the actual math behind it.”

Bell curve graphic
Figure 3: When we allow probability distributions to take complex values, their iterative convolutions behave in exotic ways never seen in classical probability theory. New mathematics must be developed in order to understand their properties.

Bui produced animations that paint a picture of the math as it evolves through convolution after convolution. These calculations occupy a realm far removed from the flat-bottomed bell curve one would see in a one-dimensional representation of a simpler problem. One of Bui’s graphics looks a bit like an expanding octopus; another resembles a melting glacier that extends into the sea below.

Bell curve graphic
Figure 4: It turns out that the seemingly unpredictable behaviors of the iterative convolution of complex probability distributions can be captured and completely understood by a collection of mathematical theorems. Bui and Randles’s most recent result is one such theorem.

The graphics opened the door to a novel theorem, detailed in their paper, that describes how these complex distributions evolve. It’s a breakthrough step in several years of work for Randles aimed at exploring the central limit theorem against ever more complex backdrops.

Currently on sabbatical in Ithaca, N.Y., Randles continues to work on further aspects of the problem he and Bui illustrated. Though the central limit theorem has endless relevance for real-world situations that require estimating probability, Randles dwells happily in the world of the theoretical.

“To me, doing mathematics is like looking at my favorite painting or listening to a song that brings me pleasure,” he said. A major focus for him as a professor is to help students gain enough fluency to be able to see math’s intrinsic beauty.

From this standpoint, he considers the project with Bui to be a great success: “Not only were we able to truly appreciate the beauty, we were able to contribute to it, too.”

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