As the distances between elements in human language sequences increase, the mutual information (MI) between them decays following a power law. This power-law relationship has been variously attributed to syntax, semantics, and discourse structure. However, the vocalizations of numerous phylogenetically distant species including humpback whales and songbirds also demonstrate similar long-range statistical dependencies. These observations support the hypothesis that long-range statistical dependencies found in human speech can occur independently of linguistic structure. To test this hypothesis, we computed MI over several child speech corpora (6 months – 12 years) to determine whether long-range relationships are present in human vocalizations prior to the production of syntactically-rich speech. We find that the developmental emergence of long-range statistical dependencies in human speech precedes the production of the linguistic structures hypothesized to generate them. This suggests that the long-range statistical organization of human speech is independent of human language syntax, semantics, or discourse structure.