Let $X_1,X_2,X_3,\ldots$ be IID r.v. with
\begin{equation} P(X_i<-1)=0 \end{equation} \begin{equation} P(X_i<0)>0 \end{equation} \begin{equation} P(X_i>0)>0. \end{equation}
Define \begin{equation} F_t = \prod_{i=1}^t(1+\frac{1}{2}X_i) \end{equation} \begin{equation} G_t = \prod_{i=1}^t(1+\frac{1}{4}X_i). \end{equation}
How can we show, for some integer $S>0$, that \begin{equation} P\left(\sup_{s\in[1,S]} \left[\sup_{t\in[1,s]} \frac{F_t-F_s}{F_t}\right] > \frac{1}{3}\right) > P\left(\sup_{s\in[1,S]} \left[\sup_{t\in[1,s]} \frac{G_t-G_s}{G_t}\right] > \frac{1}{3}\right) \end{equation}
Thanks in advance for any hints to get me started, or possibly a draft of a solution. I simply have no clue about how to proceed.
Update:
I have been thinking. Instinctvly, this problem seems to hold true because for any negative $X_i$, this holds:
\begin{equation} (1+\frac{1}{2}X_i) < (1+\frac{1}{4}X_i). \end{equation}
This should mean that $F_t$ will usually be dropping faster than $G_t$. I still wonder how to formalize these instincts.