I am working on a problem that needs to bound the increment of the absolute value of centered partial sum process and its associated renewal process. the iid partial sum process $S(t)=\sum_{i=1}^{\lfloor t\rfloor}X_i$, with $\mathbb{E}X_1 = 1/\mu>0$ and $Var[X]=\sigma^2$, $X_1>0 \ a.s.$ $$\limsup_{T\to\infty}\sup_{0\le t\le T}\sup_{0\le s\le a_T}|S(t+s)-S(t)-(t-s)/\mu|/b_T<C_1\ a.s.$$ and similarly its associated renewal process, $N(t)=\max\{k:S_k\le t\}$, $$\limsup_{T\to\infty}\sup_{0\le t\le T}\sup_{0\le s\le a_T}|N(t+s)-N(t)-\mu s|/b_T<C_2\ a.s.$$
where $a_T$ is incresing fct of $T$, $a_T/\log T\to\infty$ and $a_T\le T$.
What I want to find is $b_T=\tilde{O}(\sqrt{a_T})$. Are there any work on such prolblem? All work I can find consider the moment genertating function exists (Csorgo, Steinbach, etal). They have $b_T=\sqrt{a_T(\log T/a_T+\log\log T)}$ and for moment below $4$, they need to increase the increasing rate of $a_T$ (only allows $a_T=cT$ when only second moment exist).