The many-to-few lemma and multiple spines

TL;DR

This paper introduces a new identity that simplifies the calculation of expectations for weighted sums over particles in branching processes, extending the classical many-to-one lemma.

Contribution

It generalizes the many-to-one lemma to a many-to-few lemma, providing a more versatile tool for analyzing branching processes.

Findings

Derived a simple identity for weighted sums in branching processes

Extended the classical many-to-one lemma to a many-to-few lemma

Facilitates easier expectation calculations in complex branching models

Abstract

We develop a simple and intuitive identity for calculating expectations of weighted $k$-fold sums over particles in branching processes, generalising the well-known many-to-one lemma.