If the $(n-1)$th moment exists does the $n$th moment necessarily exist?

Question

Let's suppose the distribution is unknown but that the second moment is known to be finite. Doesn't this imply that the distribution should fall off exponentially fast and therefore higher moments must also exist?

Answer 1

No, not at all. The decay is exponential if exponential moments exist, but we cannot expect exponential decay if the second moment exists. Just consider e.g. a random variabl $X$ with density

$$f(x) := c \frac{1}{x^4} 1_{(1,\infty)}(x)$$

(where $c>0$ is chosen such that $\int f(x) \, dx =1$). Then the random variable has second moments, but $\mathbb{E}(|X|^3)= \infty$.