Past exam of the probability course of the University of Oxford / 2023 / 1 / c / iii / Solution

For $\alpha >0$, integration and independence of random variables giveBy the fractional moment criterion for convergence in probability,The parenthesized factor is less than one precisely when $\alpha \log b-\log (1+\alpha )<0$. This expression vanishes at $\alpha =0$ and has derivative $\log b-1$ there. If $b<e$, the derivative is negative, so some sufficiently small $\alpha >0$ makes the factor strictly less than one. The bound then tends to zero geometrically, proving