Concentration Inequalities for Branching Random Walk

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.