# Find choice of parameter such that a probability bound (a power function multiply an exponential function) decay exponentially with $n$

Let $$\beta:=\sqrt{\frac{n}{2\log n}}$$, consider

$$\Big (\text{Ceiling of}{\frac{\pi}{2\epsilon}}\Big)^n \cdot \exp\Big({-\frac{n-1}{\beta^2}}\Big)$$

Is it possible to find $$\epsilon>0$$, such that the expression decays exponentially with $$n$$, i.e. $$\sim O(e^{-cn})$$?

Update:

By numerical plotting, I plot the function by choosing different $$\epsilon<\pi/2$$, it seems the function always increasing. Is there any way to prove it?

• Did you try combining into a single exponential? I feel like that would probably help Commented Aug 21, 2023 at 19:35
• Are you restricting the value of $\epsilon$ ? Commented Aug 22, 2023 at 4:16
• @ClaudeLeibovici Yes. The value of $\beta$ is given as a function of $n$, so there is only $\epsilon$ left. Commented Aug 22, 2023 at 7:23
• The base of the first factor is greater than one, the second factor is essentially $1/n^2$, so it's clear that the expression is diverging for increasing $n$. As for the follow-up question: Yes, it's possible. Consider your expression with $n$ replaced by $x\in[1,\infty)$ and take the derivative. Commented Aug 31, 2023 at 13:48