We study the spiked tensor model whose noise entries follow general distributions with zero mean, unit variance, and finite fourth moment. We consider the asymmetric (independent-entry) model. Let \(s_1=\langle u^{(1)},v_0\rangle \) be the signal projection at the first iteration of the tensor power iteration method. We prove that \(\mathbb {E}[s_1]=\beta m_0^{k-1}\) and \(\textrm{Var}(s_1)=1\) for all admissible distributions of the noise entries. Furthermore, under a mild delocalization condition on the initialization of the power method, the centered projection \(s_1-\beta m_0^{k-1}\) is asymptotically Gaussian: \(s_1-\beta m_0^{k-1}\overset{d}{\rightarrow }\mathcal {N}(0,1)\) . Thus, the signal projection at the first iteration of the tensor power iteration method is asymptotically universal. We also establish concentration bounds for the projection \(s_1\) and provide a heuristic approximation formula for the normalized first iterate of the tensor power method. Finally, simulations are presented to support the theoretical results.
