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On subordinate random walks

In this article subordination of random walks in $R^d$ is considered. We prove that subordination of random walks in the sense of [BSC12] yields the same process as subordination of Lévy processes (in the sense of Bochner). Furthermore, we prove that appropriately scaled subordinate random walk converges to a multiple of a rotationally $2α$-stable process if and only if the Laplace exponent of the corresponding subordinator varies regularly at zero with index $α\in (0,1]$.