Consider a random variable on $\mathbb{R}$ that may or may not have an expectation. (E.g., if its pdf is a Cauchy distribution, it won't.) Let $p(u)$ be a probability density function on $\mathbb{R}$. Let $P(u)$ be the corresponding cumulative probability function on $\mathbb{R}$. And let $u(P)$ be the corresponding inverse cumulative probability function (or quantile function) on $[0,1)$.
I'm trying to show that, whether or not it has an expectation, a random variable can still be assigned a value using the below expression in a wide range of cases--that, whenever the pdf is continuous and eventually monotonic in both directions (non-increasing for all $u\geq n$ and non-decreasing for all $u\leq -n$ for some positive $n$), the following exists. (It's okay if it's positively or negatively infinite, just as long as it's not undefined.) $$\lim_{\varepsilon\rightarrow 0^+} \int_{\varepsilon}^{1-\varepsilon} u (P)dP $$
It's well-known that this exists if the r.v.'s expectation exists, and it equals the expectation. It's also obvious that it exists whenever the pdf is symmetric, and then equals the median. But I'm hoping that it gives us a defined value for the much broader class of pdfs that are eventually decreasing as $u$ goes to $+\infty$ and eventually increasing as $u$ goes to $-\infty$. Is it possible to show this? Or, if not, what would be a counterexample?
What I've got so far:
- If $p(u)$ is i) continuous, and ii) non-increasing for $u\geq n$ and non-decreasing for $u\leq -n$ (for some $n>0$), then it satisfies $$\lim_{z\rightarrow \infty} u\cdot p(u)=0$$
- It also satisfies $$\lim_{P\rightarrow 0^+} u(1-P)\cdot P =0\; \text{ and }\; \lim_{P\rightarrow 0^+} u(P)\cdot P=0$$
- For the above expression to be defined, it would suffice that $u(P)+u(1-P)$ is bounded either above or below on $(0,\frac{1}{2}]$.
- It would also suffice that $u(1-P(-u))-u$ is bounded both above and below. Or, equivalently, that there is some finite $M$ such that $P(z-M)<1-P(-z)<P(z+M)$ for positive $z$.
Any ideas on how to prove that the limit exists? Or ideas for how to find a counterexample?
