Clubbed Binomial Approximation for the Lightbulb Process

TL;DR

This paper introduces a novel approximation for the lightbulb process using a clubbed binomial distribution and employs Stein's method to establish bounds on the total variation distance.

Contribution

The paper develops a new clubbed binomial approximation for the lightbulb process and provides explicit bounds on the approximation error using Stein's method.

Findings

Bound on total variation distance decreases exponentially with n

Approximation accuracy improves as n increases

Method applicable to similar combinatorial processes

Abstract

In the so called lightbulb process, on days r=1,..,n, out of n lightbulbs, all initially off, exactly r bulbs selected uniformly and independent of the past have their status changed from off to on, or vice versa. With W_n the number of bulbs on at the terminal time n and C_n a suitable clubbed binomial distribution, d_{TV}(W_n,C_n) \le 2.7314 \sqrt{n} e^{-(n+1)/3} for all n \ge 1. The result is shown using Stein's method.