Can you help me to see why the following equality holds? This is in page 310 of Politis et al.(1997)
where $\{r_n\},\{b_n\}$ are nonnegative sequences, $\delta>0$ a given constant, and $U'_{n,i}$ a sum of $b_n$ random variables. At first, I thought this came from multiply and divide strategy. Does it holds only with additional assumptions?
I do not see why $(r_nb_n)^{\frac{2+\delta}{2}+\frac{2}{2+\delta}}=1$.
The $Var$-thing is exactly the same on both sides. So, we can divide by it. Then we are left with $$ \sum_{i=1}^{r_n}E|U_{n,i}'|^{2+\delta} = \frac 1{r_n^{(2+\delta)/2}}\cdot \frac{1}{r_n^{2/(2+\delta)}b_n^{2/(2+\delta)}}\sum_{i=1}^{r_n}E|b_n^{-1/2}U_{n,i}'|^{2+\delta}. $$ Now, $E(\cdot)$ is a linear function, so you can pull $(b_n^{-1/2})^{2+\delta} = 1/b_n^{(2+\delta)/2}$ out of it (I assume that $b_n$ is not random). So, we can divide by the sum and get $$ 1 = \frac 1{r_n^{(2+\delta)/2}}\cdot \frac{1}{r_n^{2/(2+\delta)}b_n^{2/(2+\delta)}}\cdot\frac 1{b_n^{(2+\delta)/2}} = \frac 1{(r_nb_n)^{\frac{2+\delta}2 + \frac 2{2+\delta}}}, $$ which is obviously false. I guess the guys on the paper had some problems with power laws.