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 during which 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 2 January 1990 to 17 October 2025, we construct their associated returns to obtain a non-arbitrary sample of multi-scale returns, which we call trend returns (TReturns). The timescale of each multi-scale return is determined by the exponentially distributed duration of its corresponding run. We empirically show that the distribution of these coarse-grained returns exhibits distinctive statistical properties: the central region displays an exponential decay, likely resulting from the exponential distribution of trend durations, 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 additional evidence of its natural emergence. We also explore the informational properties 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 suppressed randomness. Overall, these findings suggest that constructing TReturns filters out microscopic noise, reveals structured temporal patterns, and provides a complementary and clear view of market behavior.