Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$.
For $n\in \mathbb{N}$ the sequence $K_{n}=\sup_{t\in [n,n+1]}\left|X_t-X_n\right|$ is i.i.d.
Can we conclude, that $\sqrt{n}^{-1}K_{n}\rightarrow 0$ a.s.?
If not, by centering $\tilde{X}_t=X_t-E\left[X_t\right]$ is a martingale. Defining $\tilde{K}_{n}$ analogous via $\tilde{X}_t$, can we conclude $\sqrt{n}^{-1}\tilde{K}_{n}\rightarrow 0$ a.s.?
Note in both cases $X_t$ and $\tilde{X}_t$ can be written as random walks, for which one can use inequalities e.g. Martingale maximal inequality.
I hope you can help me out. Best regards