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de Finetti's theorem roughly states that infinite sequence of exchangeable random variables are conditionally independent. I am looking for tight bounds for de Finetti's theorem in the following scena …

I think it maybe possible to use a martingale concentration inequality for this application. Indeed, let $f(X_1, \cdots, X_n) = \sum_{m} X_m^2$. Then $F_i = \mathbb{E}[f|X_1, \cdots, X_i]$ is a doob m …

Chebyshev's sum inequality can be proven more concisely using probability. Indeed, if $f$ and $g$ are two monotone functions and $X$ a random variable, then we have $$ (f(X_1)-f(X_2))(g(X_1)-g(X_2)) \ …

It is known that the max of i.i.d. subgaussian random variables with variance $\sigma^2$ is on the order of $\sigma \sqrt{\log n}$ so you can expect the squared max of the random variables to be rough …

Let $X_1, X_2, \ldots$ be independent and identically distributed random variables with mean $0$ and variance $1$ and let $S_n = (X_1 + \cdots + X_n)/\sqrt{n}$ to be their normalized sum. Define $D_n$ …