# A central limit theorem for the gossip process

## TL;DR

This paper proves a central limit theorem for the Aldous gossip process, showing that the random time shift in information dissemination is approximately normally distributed, with computable mean and variance.

## Contribution

It extends the understanding of the gossip process by establishing a normal approximation for the initial stochastic delay, enhancing predictive accuracy.

## Key findings

- The random time shift follows an approximately normal distribution.
- The mean and variance of the time shift can be explicitly computed.
- The broad deterministic description remains valid with increased precision.

## Abstract

The Aldous gossip process represents the dissemination of information in geographical space as a process of locally deterministic spread, augmented by random long range transmissions. Starting from a single initially informed individual, the proportion of individuals informed follows an almost deterministic path, but for a random time shift, caused by the stochastic behaviour in the very early stages of development. In this paper, it is shown that, even with the extra information available after a substantial development time, this broad description remains accurate to first order. However, the precision of the prediction is now much greater, and the random time shift is shown to have an approximately normal distribution, with mean and variance that can be computed from the current state of the process.