We introduce the stable sums method for inference of extreme return levels for multivariate stationary time series. This new method is based on large deviation principles for regularly varying time series which allows for incorporation of time and space extreme dependencies in the analysis. It avoids classical declustering steps as implementation coincides for both independent and dependent observations from the stationary model. A comparison with the main estimators from extreme value theory, where detecting clustering in time is required, shows improvement of the coverage probabilities of confidence intervals obtained from our method against its competitors. Numerical experiments also point to a smaller mean squared error with the multivariate stable sums method compared to an univariate approach. We apply our method for inference of high return levels of multivariate daily fall precipitation measurements in France.