The presented methodology for testing the goodness-of-fit of an Autoregressive Hilbertian model (ARH(1) model) provides an infinite-dimensional formulation of the approach proposed in Koul and Stute (1999), based on empirical process marked by residuals. Applying a central and functional central limit result for Hilbert-valued martingale difference sequences, the Wiener-type limiting process of the H-valued empirical process indexed by an H-valued covariate is obtained. The performance of the proposed testing procedure is illustrated by simulations. Consistency of this asymptotically distributed free test is derived. Since, in practice, the autocorrelation and autocovariance operators of the ARH(1) process are unknown, the uniform asymptotic equivalence in probability in H-norm of the test statistics under totally and misspecified scenarios is established.

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