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The Berry-Essen bound stated that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$ where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...

Let $X_{1}, \ldots, X_{T} \in \{0, 1\}$ be a sequence of Boolean random variables with $$ \mathbb{E}[X_{t} | X_{1}, \dots, X_{t - 1}] = p_{t}. $$ Consider the sequence $Y_{t} := X_{t} - p_{t}$ (which ...

Are there any easy to use bounds on sums of the following kind : $$ \sum_{i = 1}^{i = N} |a_i| \geq P \\ a_i \sim \mathcal{N}(0, 1) \\ $$ and also for sums of the form : $$ \sum_{i = 1}^{i = M} \...