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 $J_{\mathrm{LZ}}=min\{{\tilde{C}}_{\mathrm{LZ}}\left(x\right),{\tilde{C}}_{\mathrm{LZ}}\left(y\right)\}$ is a calibrated diagnostic, not a causal test. On two toy models—coupled logistic maps and multivariate fractional Brownian motion (${dim}_H=2-H$)—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 $J_{\mathrm{LZ}}$ alongside $\rho$.