Concentration Inequalities for Branching Random Walk
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917289818849280 |
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| author | Liu, Changqing |
| author_facet | Liu, Changqing |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_05860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration Inequalities for Branching Random Walk Liu, Changqing Probability 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. |
| title | Concentration Inequalities for Branching Random Walk |
| topic | Probability |
| url | https://arxiv.org/abs/2509.05860 |