Statistical Distribution and Entropy of Multi-Scale Returns: A Coarse-Grained Analysis and Evidence for a New Stylized Fact

Financial time series often show periods where market index values or asset prices increase or decrease monotonically.These events are known as price runs, uninterrupted trends, or simply runs. By identifying such runs in the daily DJIA and IPC indices from 01/02/1990 to 10/17/2025, we construct their associated returns, to obtain a non-arbitrary sample of multi-scale returns, we named trend returns (TReturns). The time scale for each multi-scale return is determined by the exponentially distributed duration of its respective run. We empirically reveal that the distribution of these coarse-grained returns show interesting statistical properties: the central region displays an exponential decay, likely resulting from the exponential trend duration, while the tails follow a power-law decay. This combination of exponential central behavior and asymptotic power-law decay has also been observed in other complex systems; and our findings provide an additional evidence of its natural emergence. We also explore the informational aspects of multi-scale returns using three measures: Shannon entropy, permutation entropy and compression-based complexity. We find that Shannon entropy increases with coarse-graining, indicating a wider range of values; permutation entropy drops sharply, revealing underlying temporal patterns and compression ratios improve, reflecting suppresed randomness. Overall, these findings suggest that constructing TReturns filters out microscopic noise, reveals structureded temporal patterns, and provides a complementary and clear view of market behavior.