Let $\rho:\mathbb R^d\to\mathbb R_+$ be a density function with finite first moment, i.e.
$$\int_{\mathbb R^d}~ \rho(x)dx~=~1 \quad \mbox{ and }\quad \int_{\mathbb R^d}~ |x|\rho(x)dx<+\infty.$$
For each $n\ge 1$ and $q=(q_1,\ldots, q_d)\in\mathbb Z^d$, define the box $V_q^n$ by
$$V_q^n \quad :=\quad \left\{x=(x_1,\ldots, x_d):\quad \frac{q_i}{n} ~\le~ x_i ~<~ \frac{q_i+1}{n},\quad \mbox{for } i=1,\ldots, d \right\}.$$
Let
$$I ~:=~ \int_{\mathbb R^d} |x|\rho(x)dx \quad \mbox{and} \quad I_n~ :=~ \sum_{q\in\mathbb Z^d} \int_{V_q^n} \frac{|q|}{n}\rho\left(\frac{q}{n}\right)dx ~ = ~ \sum_{q\in\mathbb Z^d} \frac{|q|}{n^{d+1}}\rho\left(\frac{q}{n}\right).$$
I am unable to find references to estimate $|I_n-I|$ (under suitable conditions), even it seems to be a very elementary question. Any comments or references will be very much appreciated! Thanks a lot!
