Summary: Schmidt's Tauberian theorem says that if a sequence \((x_k)\) of real numbers is slowly decreasing and \(\lim_{n\to \infty} (1/n) \sum^n_{k=1} x_k = L\), then \(\lim_{k\to\infty} x_k = L\). The notion of slow decrease includes Hardy's two-sided as well as Landau's one-sided Tauberian conditions as special cases. We show that ordinary summability \((C,1)\) can be replaced by the weaker assumption of statistical summability \((C,1)\) in \textit{R. Schmidt}'s theorem [see Math. Z. 22, 89--152 (1925; JFM 51.0182.04)]. Two recent theorems of \textit{J. A. Fridy} and \textit{M. K. Khan} [see Proc. Am. Math. Soc. 128, 2347--2355 (2000; Zbl 0939.40002)] are also corollaries of our Theorems 1 and~2. In the appendix, we present a new proof of \textit{T. Vijayaraghavan}'s lemma [J. Lond. Math. Soc. 1, 113--120 (1926; JFM 52.0221.01)] under less restrictive conditions, which may be useful in other contexts.