Stationarity, Temporal Dependence, and Shuffling Time Series

Summary

The discussion distinguishes stationarity from the presence of temporal structure. A stationary Ornstein–Uhlenbeck process can still exhibit mean reversion: when its value moves far from its long-run level, that deviation provides information about its likely future movement. Stationarity therefore does not imply that observations are independent or that trends and other dependencies cannot be modeled.

The answer contrasts this with a martingale, where past information does not improve prediction of the next directional move over a fair-coin baseline. It notes that shuffling increments of a Brownian motion can produce another realization of that process. This illustrates why the relevance of shuffling depends on the process and on what is being shuffled. The brief exchange offers conceptual examples rather than a general test for stationarity or a method for modeling market regimes; its claims about martingales concern directional predictability, not every possible feature of a time series.

Key ideas

• A stationary process can retain temporal dependence, as mean reversion in an Ornstein–Uhlenbeck process illustrates.
• A martingale has no past-information advantage for predicting the next directional move beyond a fair-coin baseline.
• Shuffling Brownian-motion increments can produce another realization of the process.
• Stationarity alone does not establish whether a series contains exploitable predictive structure.

Tags

# Temporal dependencies in time-series


# Temporal dependencies in time-series












To my knowledge, the algorithms that require stationary input can't capture temporal dependencies. This is inherent due to the fact that the input features must be stationary, thus things like trends, changing market regimes, etc can't be effectively input into the model.

Therefore, when plugging in features to this model, it theoretically shouldn't matter if I were to randomize the time-series before AND THEN calculate these stationary features (like RSI, for example).

Please correct me if I'm wrong, just something that I was thinking about.

## Answer by user3072048 (score 1)

https://quant.stackexchange.com/a/81386

Take an Ornstein-Uhlenbeck for example. The process becomes weakly stationary for large times. Yet, it has temporal dependencies: Whenever the spot is far away from its long-term average (mean-reversion level), it will revert back to it. This is a temporal structure.

The situation is different for time-series where the underlying process is a martingale (fair game). Such a series is non-stationary and has no temporal structure in the sense that any information available at a given time $t$ would allow you to predict the next directional move better than a fair coin flip.

Hence it is rather the other way around as you suggest: In a Brownian motion for example that is simplest martingale process, you could shuffle the increments and create another 'realization' of a Brownian motion this way

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