Exercise 2.20: Tail Bounds Under Moment Conditions
Starting from Markov’s inequality \begin{align} \P \biggl[ \biggl|\sum_{i=1}^n X_i \biggr| \geq n \delta \biggr] &\leq (n \delta)^{-2m} \E \biggl[ \biggl( \sum_{i=1}^n X_i \biggr)^{2m} \biggr] \newline &\overset{\text{(i)}}{\leq} (n \delta)^{-2m} R_m \left[ \sum_{i=1}^n \E [X_i^{2m}] + \biggl( \sum_{i=1}^n \E [X_i^2] \biggr)^m \right] \newline &\overset{\text{(ii)}}{\leq} (n \delta)^{-2m} R_m (n C_m^{2m} + n^m C_m^{2m}) \newline &\leq 2 R_m \biggl( \frac{C_m}{\sqrt{n} \delta} \biggr)^{2m} \, , \end{align} where (i) is by the Rosenthal’s inequality provided in the hint, and (ii) follows by applying the assumed moment bound.