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Let $(g_n)_{n\geq 1}$ be a sequence of independent and identicallydistributed 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 provethe strong and weak laws of large numbers and the central limit theorem for$\langle f, G_n v \rangle$, which improve the previous results establishedunder the exponential moment condition on $\mu$. We further demonstrate theBerry-Esseen bound, the Edgeworth expansion, the Cram\'{e}r type moderatedeviation expansion and the local limit theorem with moderate deviations for$\langle f, G_n v \rangle$ under the exponential moment condition. Under asubexponential moment condition on $\mu$, we also show a Berry-Esseen typebound and the moderate deviation principle for $\langle f, G_n v \rangle$. Ourapproach is based on various versions of the H\"older regularity of theinvariant measure of the Markov chain $G_n \!\cdot \! x = \mathbb R G_n v$ onthe projective space of $V$ with the starting point $x = \mathbb R v$.

Limit theorems for the coefficients of random walks on the general linear group