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## Abstract for "Testing the stability of the functional autoregressive process" by Piotr Kokoszka

The functional autoregressive process has become a useful tool in the analysis of functional time series data. It is defined by the equation $X_{n+1}= \Psi X_n + \eg_{n+1}$, in which the observations $X_n$ and errors $\eg_n$ are curves, and $\Psi$ is an operator. To ensure meaningful inference and prediction based on this model, it is important to verify that the operator $\Psi$ does not change with time. We propose a method for testing the constancy of $\Psi$ against a change--point alternative which uses the functional principal component analysis. The test statistic is constructed to have a well--known asymptotic distribution, but the asymptotic justification of the procedure is very delicate. We develop a new truncation approach which together with Mensov's inequality can be used in other problems of functional time series analysis. The estimation of the principal components introduces asymptotically non-negligible terms, which however cancel because of the special form of our test statistic (CUSUM type). The test is implemented using the R package fda, and its finite sample performance is examined by application to credit card transaction data.