The paper can be found at this link: http://www.columbia.edu/~sk75/sinica.pdf. The proof is not given but there is a reference to the paper: https://link.springer.com/article/10.1007/BF00531747 which has a proof but the conclusions do not match and I cannot figure out a connection. So I need either to make such a connection. Or prove the following theorem myself.
In the first paper, the most important theorem is proved solely based on the theorem 3.1 (on which I need help proving) which states
Theorem: For any constants $b\geq y$ and $b>0$, as $m\rightarrow\infty$, $$\mathbb{P}\Big(\ U_{m}<y\sqrt{m}\ ,\ \tau'(b,U)\leq m\ \Big)\\ =\mathbb{P}\Big(\ U(1)\leq y\ ,\ \tau(b+\beta/\sqrt{m},U)\leq1\ \Big)+o(m^{-1/2})$$ where the constant $\beta=-(\zeta(\frac{1}{2})/\sqrt{2\pi})$. Where $\zeta$ is the Riemann zeta function.
Here:
- $\tau'(b,U):=\inf\{n\geq1:U_{n}\geq b\sqrt{m}\}$,
- $\tau(b,U):=\inf\{t\geq0:U(t)\geq b\}$ and
- $U_{n}=\sum_{i=1}^{n}\left(Z_{i}+\frac{v}{\sqrt{m}}\right)$ where $Z_i$ is a standard random variable and $U(t):=vt+B(t)$ where $B(t)$ is the standard Brownian motion.
I really don't know how to prove this and my supervisor is really busy. So I would really appreciate some guidance or just a complete proof.
Thank you so much!