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Abstract

Classical concentration inequalities for random walks are typically derived under independence assumptions or martingale difference conditions. In this paper, we extend concentration inequalities to more general conditions, relaxing the classical assumptions of independence and martingale differences. Specifically, in the notation of calculus, while the condition of independence is $\displaystyle \frac { \partial y } {\partial t}=$ constant and martingale difference condition is $\displaystyle \frac { \partial y } {\partial t} = 0 $, we relax these conditions to $\displaystyle \frac { \partial^2 y } {\partial u_i \partial t} \le L$, thereby allowing $\displaystyle\frac { \partial y } {\partial t}$ to vary. Furthermore, concentration inequalities are established for branching random walk.