Time Series Correlations and Kolmogorov Complexity: A Hausdorff Dimension Perspective

Spurious correlations between time series are a persistent problem: simple, low-complexity patterns are abundant, so unrelated series can easily exhibit high Pearson correlation. We argue that Kolmogorov complexity-a series' resistance to compression-provides a principled diagnostic for flagging such cases. We prove an algorithmic trilemma: a pair of binary sequences cannot simultaneously be algorithmically independent, highly correlated, and highly complex. This gives a deterministic complexity ceiling for independent correlated pairs and a probabilistic bound under which spurious correlations among independent high-complexity pairs are exponentially rare; we further bridge these results to an effective Hausdorff dimension obstruction. These guarantees hold for binary sequences under Hamming correlation; their extension to real-valued series via serialisation and LZ compression is empirically validated rather than proved, so the joint indicator JLZ=min{C˜LZ(x),C˜LZ(y)} is a calibrated diagnostic, not a causal test. On two toy models-coupled logistic maps and multivariate fractional Brownian motion (dimH=2-H)-false positives are far more common among low-complexity series. Because noise inflates complexity and non-stationary processes can be both complex and spuriously correlated, we recommend a two-stage workflow: establish stationarity, then report JLZ alongside ρ.