Concentration Inequalities for Branching Random Walk

TL;DR

This work broadens the scope of concentration inequalities beyond independence and martingale-difference assumptions by introducing a calculus-based framework in which the drift term $\partial y/\partial t$ can depend on past state, controlled by a uniform bound on the mixed derivative $|\partial^2 y/(\partial u_i \partial t)|$. It then extends these ideas to branching random walks (BRW), proving concentration under negative association and exponential moment conditions and deriving a BRW-analogue of Azuma-type bounds using BRW-weighted expectations $E^{(M)}$. The results connect to ongoing methods such as the differential-equation method and have implications for complex combinatorial problems like K-SAT and q-COL (with a detailed treatment in a companion paper). Overall, the paper provides a calculus-based perspective on stochastic concentration and a rigorous framework for concentration in BRW contexts with dependent dynamics.

Abstract

While classical concentration inequalities are typically restricted to two special cases -- independence and martingale difference sequences -- we extend concentration inequalities to a much broader class of stochastic processes by relaxing these foundational conditions. %\vspace{0.2\baselineskip} Specifically, heuristically and in the language of calculus, while independence and the martingale difference property correspond to \[ \displaystyle \frac { \partial y } {\partial t}= \text{constant}, \quad \displaystyle \frac { \partial y } {\partial t} = 0 \] respectively, %\vspace{0.3\baselineskip} we relax these conditions to %\[ \left| \frac { \partial^2 y } {\partial u_i \, \partial t} \right| \le L, \] %thereby allowing the drift $\displaystyle\frac { \partial y } {\partial t}$ to vary with past state $u_i$. \vspace{0.3\baselineskip} a general setting that requires only the existence of a drift $\displaystyle\frac { \partial y } {\partial t}$ which is allowed to vary with the past state. \vspace{0.3\baselineskip} Furthermore, concentration inequalities are established for branching random walks.

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Theorem 2.1
• proof : Proof (Downgraded version of Azuma's inequality)
• Lemma 2.2
• proof : Proof (Lemma \ref{['Lip_f']})
• Lemma 2.3
• proof : Proof of Lemma \ref{['con_extension']}
• Theorem 2.2
• proof