Suppose that the properly normalized partial sums of a sequence of independent identically distributed random variables with values in a separable Banach space converge in distribution to a stable law of index $\alpha$. Then without changing its distribution, one can redefine the sequence on a new probability space such that these partial sums converge in probability and consequently even in $L^p (p < \alpha)$ to the corresponding stable process. This provides a new method to prove functional central limit theorems and related results. A similar theorem holds for stationary $\phi$-mixing sequences of random variables.
"Weak and $L^p$-Invariance Principles for Sums of $B$-Valued Random Variables." Ann. Probab. 8 (1) 68 - 82, February, 1980. https://doi.org/10.1214/aop/1176994825