Central-limit convergence is only in distribution

The conclusions of Lindeberg-Levy iid central limit theorem, Lindeberg-Feller central limit theorem: sufficiency and Multivariate iid central limit theorem compare the row-sum laws with a Gaussian law. In Convergence in distribution of random elements, the limit is specified by tests on laws; the theorem does not construct a Gaussian random vector jointly with the original summands. The AC hypotheses of the cited theorems remain in force when applying them.

By contrast, Convergence in probability requires the probabilities of distance-from-the-limit events on a common probability space to tend to zero. Almost-sure convergence of real random variables requires pointwise convergence outside a null set on such a space. Neither conclusion is supplied merely by identifying the Gaussian limit law; a coupling or further argument is required. These statements do not deny that a suitable coupling can sometimes give stronger convergence.

The special result Convergence in distribution to a constant is convergence in probability concerns a point-mass limit. It applies to scalar degenerate normal limits on a common space, but does not turn a positive-variance normal limit into convergence in probability to a newly chosen Gaussian. A multivariate fully degenerate Gaussian limit requires the corresponding vector argument, which that real-valued theorem does not supply; a singular covariance with some positive-variance directions still gives a nonconstant law. No convergence rate or almost-sure version is asserted here.