Copulas are useful for studying how several random variables depend on each other, because they allow the dependence structure to be separated from the individual marginal distributions. This makes them particularly useful when the main interest is in whether the dependence itself changes over time. In this thesis, I study this problem for a multivariate time series whose dependence is described by a copula from a fixed parametric family. The main goal is to test whether the copula parameter remains constant over the whole sample or changes once at an unknown point in time. For each possible change-point, a model with one common copula parameter is compared with a model that allows different parameters before and after the split. Since the change location is unknown, the final test statistic is obtained by taking the largest value over a trimmed set of candidate change-points.

I first consider the ideal case, where the marginal distributions are known. In this setting, the true copula observations are available directly. The main difficulty is that the test does not use only one fixed split point, but searches over many possible locations. Therefore, the likelihood approximations need to hold uniformly over all candidate change-points. To do this, I first control the score process and the observed information matrices uniformly over the candidate splits. These results are then used to describe the estimation errors of the maximum likelihood estimators. From there, the likelihood ratio statistic can be rewritten as a quadratic form involving a centered score process. This process converges to a standard multivariate Brownian bridge, which gives the limiting distribution of the maximized test statistic.

The feasible case is more difficult because the marginal distributions are unknown. The true copula observations are then replaced by rank-based pseudo-observations. For every possible change-point, the ranks are recomputed separately in the two resulting segments. Because of this, the score terms are no longer simple partial sums of independent observations. To handle this, I use the sequential localrank empirical copula process and combine it with a function-indexed integration-by-parts argument. This makes it possible to derive the weak limit of the local-rank score process and, from this, the limiting distribution of the feasible likelihood ratio statistic. The resulting limit still has a Brownian-bridge-type structure, but its covariance also reflects the additional effect of estimating the marginal distributions through ranks.

Finally, I study the finite-sample behaviour of the test through simulations using Gaussian and Student 𝑡 copulas. The results show a clear general pattern: the test becomes more powerful when the sample size increases or when the change in dependence becomes larger. Small changes are harder to detect, especially for the feasible procedure based on local ranks. At the same time, the difference between the ideal and feasible procedures becomes smaller for larger samples and stronger changes in dependence. Overall, the thesis develops the theoretical justification for a likelihood-ratio change-point test in copula models and studies how the procedure behaves in finite samples.

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