In Boukanjime et al. " https://www.sciencedirect.com/science/article/pii/S0005109821004039 " I'm having difficulties understanding the proof of Theorem 5.1 after equation (13). It's a relatively short proof and reading the entire paper is not required.
- I don't understand after how the limits of the supremum changed after (13)?
- I don't understand how they applied the Borel-Cantelli lemma after the result of using Chebyshev's inequality?
- How did they go from $$\sup_{k\leq t\leq k+1 }\tilde{V}_1(X(t))\leq k^{1+\epsilon}$$ to$$\frac{\log \tilde{V}_1(X(t))}{\log t}\leq \frac{(1+\epsilon)\log k}{\log k}.$$
Of course they applied $\log$ to both sides from the first inequality, but then how did two different denominators appear?
