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4 votes
1 answer
189 views

Sign of error in the central limit theorem

Let $X_n$ and $Y_n$ be independent copies of two random variables $X$ and $Y$ with domain $\{-1,0,1\}$ for $n\in \mathbb{N}$. For a given $k\in \mathbb{N}$, I would like to find conditions on $X$ and $...
Flo Dorner's user avatar
1 vote
1 answer
285 views

Rate of convergence to uniform distribution

Let $p=(p(1),\ldots,p(N))$ be a discrete distribution on $[N]:=\{1,2,\ldots,N\}$ with full support (i.e all the $p(i)$'s are strictly positive and sum to $1$). Let $i_1,i_2,\ldots,i_T$ be an iid ...
dohmatob's user avatar
  • 6,853
1 vote
0 answers
96 views

Concentration for $\sum_{i=1}^n y_i \psi(x_i^\top u)$, for $y_1,\ldots,y_n \sim \{\pm 1\}$ and $x_1,\ldots,x_n$ uniform iid on hypersphere

Let $y_1,\ldots,y_n$ be drawn iid uniformly from $\{\pm 1\}$ and let $x_1,\ldots,x_n$ be drawn iid uniformly from the unit-sphere $(d-1)$-dimensional sphere $\mathbb S_{d-1}$, and independently from ...
dohmatob's user avatar
  • 6,853
9 votes
2 answers
1k views

Adaptive version of the Azuma–Hoeffding inequality

The Azuma inequality states that if we have a martingale $X_1,\ldots,X_N$ that satisfies a bounded difference condition: $$|X_k - X_{k-1}| \leq c_k$$ Then: $$\Pr\left[X_N - X_0 \geq \sqrt{2\sum_kc_k^2 ...
Aaron's user avatar
  • 794