$^{75}$As nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) measurements have been carried out to investigate the magnetic and electronic properties of the filled skutterudite metallic compound SrFe$_4$As$_{12}$. The temperature dependence of Knight shift $K$ determined by the NQR spectrum under a small magnetic field ($le$ 0.5 T) shows the similar temperature dependence of the magnetic susceptibility $chi$ which exhibits a broad maximum at $T^ast$ $sim$ 50 K. The nuclear spin-lattice relaxation rate divided by temperature, 1/$T_1T$, increases with decreasing temperature and exhibits a broad maximum at $T$ $sim$ 70 K, similar to the case of $chi$. The temperature dependence of $K$ and $1/T_1T$ is reasonably explained by a simple model where we assume a concave-shaped band structure near the Fermi energy. Based on a Korringa ratio analysis using the $T_1$ and $K$ data, ferromagnetic spin fluctuations are found to exist in SrFe$_4$As$_{12}$. These results indicate that SrFe$_4$As$_{12}$ can be characterized to be a metal with ferromagnetic correlations and also the peculiar band structure responsible for the suppression of $1/T_1T$ and $K$ at low temperatures.
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