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Jun 21 at 10:25 comment added Joseph Basford Thanks for your answer (+1). In the link provided by the question poster, it is stated that the tail bound only applies when $s\geq 1$. This is the source of the $\delta_n^2$ term in the expectation bound, as my answer (hopefully) makes clear.
Jun 20 at 18:26 history answered Adrian Keister CC BY-SA 4.0