In recent years, the Kalman filter has become the prime approach for estimating parameters that evolve following some dynamic model and prior statistics. In addition, recent contributions are introducing the use of autoregressive models in the state-space formulation to deal with correlated Gaussian-distributed magnitudes. However, the derivation of closed-form expressions for predicting their performance during the design stage is still an open problem. In that regard, in this letter we derive novel approximate closed-form upper bounds to characterize the convergence time of autoregressive Kalman filters. To this end, we extend a batch mode-based approach previously proposed in the literature that reveals the need for a dedicated dual-asymptotic analysis for this kind of techniques. Simulations are provided to show the goodness of the derived results.