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Differentially private wireless federated learning (DPWFL) is a promising framework for protecting sensitive user data. However, foundational questions on how to precisely characterize privacy loss remain open, and existing work is further limited by convergence analyses that rely on restrictive convexity assumptions or ignore the effect of gradient clipping. To overcome these issues, we present a comprehensive analysis of privacy and convergence for DPWFL with general smooth non-convex loss objectives. Our analysis explicitly incorporates both device selection and mini-batch sampling, and shows that the privacy loss can converge to a constant rather than diverge with the number of iterations. Moreover, we establish convergence guarantees with gradient clipping and derive an explicit privacy-utility trade-off. Numerical results validate our theoretical findings.