An almost sure upper bound for random multiplicative functions on integers with a large prime factor

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\epsilon >0$ small. We prove that, as $x\to +\mathrm{\infty }$, we almost surely have $$\mid \sum _{\begin{array}{c}n\le x\\ P(n)>\sqrt{x}\end{array}}f(n)\mid \le \sqrt{x}(\log \log x{)}^{1\mathrm{/}4+\epsilon },$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.