bounds of expectation

Question

I have a situation where I have one to one correspondence between $\mathbb N$ and a subset of probability measures on $\{1,2,3,\dots, m\}$ i.e $n\mapsto p_n$, Could anyone tell me, what is the worst possible case I can expect in terms of the expectation of these changes? Mathematically: May I expect some bounds on $\mathbb E(P_{N}-P_{N+1})$?

Answer 1

Defining $$ \|P-Q\|_{\text{TV}} = \frac 1 2 \sum_{x=1}^m |p(x)-q(x)|,$$ where $p(x)$ means $P(\{x\})$ and so on, you have the simple inequality \begin{align*} &\left|\sum_{x=1}^m x \big(p(x)-q(x)\big)\right |\\ =& \left|\sum_{x=1}^m x\big(p(x)-q(x)\big)-\frac{(m+1)}2\sum_{x=1}^m\big(p(x)-q(x)\big)\right |\\ =& \left|\sum_{x=1}^m (x-\frac{m+1}2)\big(p(x)-q(x)\big)\right |\\ \le& \frac{m-1}2\sum_{x=1}^m |p(x)-q(x)|\\ =& (m-1)\|P-Q\|_{\text{TV}}\end{align*}