Boundaries on the single-period returns

I know that $$e^{t\mu_{\operatorname{log}}-\Gamma\sqrt{t}\sigma_{\operatorname{log}}}\leq \widetilde{R}_t^S \leq e^{t\mu_{\operatorname{log}}+\Gamma\sqrt{t}\sigma_{\operatorname{log}}}$$, with $$\mu_{\operatorname{log}},\sigma_{\operatorname{log}},\Gamma \in \mathbb{R}^+$$ and $$\widetilde{R}_t^S:=\prod_{i=1}^{t-1}(1+\widetilde{r}_t^S)$$. Note that this interval results from $$\left | \frac{\frac{1}{t}\operatorname{log}\widetilde{R}_t^S-\mu_{\operatorname{log}}}{\frac{\sigma_{\operatorname{log}}}{\sqrt{t}}} \right | \leq \Gamma$$, in its turn obtained from the Central Limit Theorem $$\left | \frac{\frac{1}{n}\sum_{i=1}^{n}X_i-\mu}{\frac{\sigma}{\sqrt{n}}} \right |\overset{d}{\rightarrow} \operatorname{N}(0,1)$$. How can I prove that the interval for $$\widetilde{r}_t^S$$ is

$$\left | \widetilde{r}_t^S-\mu \right |\leq \Gamma\sigma \space \Rightarrow \space \mu-\Gamma\sigma \leq \widetilde{r}_t^S \leq \mu+\Gamma\sigma$$

knowing that $$\frac{\widetilde{R}_{t+1}^S}{\widetilde{R}_t^S}=1+\widetilde{r}_t^S$$?

Thanks for any help.

• Probably using a Taylor expansion of the exponent or log... Nov 24, 2020 at 17:31
• @steveo'america That is to say? Nov 25, 2020 at 8:48