  Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically
distributed random elements with law $\mu$ on the general linear group
$\textrm{GL}(V)$, where $V=\mathbb R^d$. Consider the random walk $G_n : = g_n
\ldots g_1$, $n \geq 1$, and the coefficients $\langle f, G_n v \rangle$, where
$v \in V$ and $f \in V^*$. Under suitable moment assumptions on $\mu$, we prove
the strong and weak laws of large numbers and the central limit theorem for
$\langle f, G_n v \rangle$, which improve the previous results established
under the exponential moment condition on $\mu$. We further demonstrate the
Berry-Esseen bound, the Edgeworth expansion, the Cram\'{e}r type moderate
deviation expansion and the local limit theorem with moderate deviations for
$\langle f, G_n v \rangle$ under the exponential moment condition. Under a
subexponential moment condition on $\mu$, we also show a Berry-Esseen type
bound and the moderate deviation principle for $\langle f, G_n v \rangle$. Our
approach is based on various versions of the H\"older regularity of the
invariant measure of the Markov chain $G_n \!\cdot \! x = \mathbb R G_n v$ on
the projective space of $V$ with the starting point $x = \mathbb R v$.
