Let $\rho:\mathbb R^d\to\mathbb R_+$ be a density function with finite first moment, i.e.
$$\int_{\mathbb R^d}~ \rho(x)dx\quad = \quad 1,\quad\quad\quad \int_{\mathbb R^d}~ |x|\rho(x)dx\quad < \quad +\infty.$$
For each $q=(q_1,\ldots, q_d)\in \mathbb Z^d$, let $V_q\subset \mathbb R^d$ be defined by $V_q:= \Pi_{i=1}^d~ \left[q_i, q_i+1\right)$. My question is that, under what conditions on $\rho$, one has an estimation on the tail given by
$$\sum_{q:~ |q|\ge n}~ |q| \rho(q) \lambda(V_q),\quad \quad \quad (\ast)$$
where $|q|:=\sum_{i=1}^d q_i$ and $\lambda(V_q)$ denotes the Lebesgue measures of $V_q$. Thanks a lot!
PS: Indeed, the objective is to estimate the following term:
$$\sum_{q\in\mathbb Z^d}\int_{V_q}~ \big(|x|\rho(x)-|q|\rho(q)\big)dx.$$
Clearly a necessary condition is to assume that $\rho$ is continuous. However, to ensure the asymptotic behaviour for large $|q|$, the estimation of $(\ast)$ is needed.