Let $F=\{g_t\}$ be a one-parametr diagonal subgroup of $SL_n(\mathbb R)$.

We assume  $F$ has no nonzero invariant vectors in $\mathbb R^n$.

Let $x\in X, \varphi\in C_c(X)$ and $\mu$ be the probability Haar measure

on $X$. For certain proper subgroup $U$ of the unstable horospherical  subgroup

of $g_1$ we show that for almost every $u\in U$

\[

\frac{1}{T}\int_0^T\varphi({g_tux})dt \to \int_X\varphi d\mu.

\]

If $\varphi$ is moreover smooth, we can get an  error rate of the convergence.

The error rate is ineffective due to the use of Borel-Cantelli lemma.