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In Hoeffding's inequality $$ P(S_n-E[S_n]\geq t)\leq \exp(\frac{t^2}{\sum_{i=1}^n(a_i-b_i)^2}), $$ where $S_n=X_1+X_2+\dots+X_n$, with $X_i$ independent random variables satisfying $a_i<X_i<b_i$, is it allowed to let $a_i$ and $b_i$ depend on $n$?

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Yes, it is allowed. Note that the inequality is a statement for fixed $n$, hence there is no problem with $a_i$ and $b_i$ depending on $n$.

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This is like asking if the $a_i$'s in the set $\{a_1, a_2, ..., a_n\}$ are allowed to depend on $n$. They can. Consider $a_i = i/n$ like in Riemann sums. Then

$$\{a_1, a_2, ..., a_n\} = \{1/n, 2/n, ..., n/n\}$$

We can have $a_i = (i-1)/n$

Then

$$\{a_1, a_2, ..., a_n\} = \{0/n, 1/n, ..., (n-1)/n\}$$

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