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This paper uses a functional approach to analyze the distributions of weekly returns in Bitcoins on leading cryptocurrency exchanges. The results present strong evidence for non-stationarity, which suggests unpredictability and time-varying statistical properties. In addition, non-stationary fluctuations tend to be primarily concentrated in even moments, such as volatility and kurtosis—however their effect is significant and persistent in every moment, including higher moments. The analysis in this paper proposes that the Bitcoin market is maturing and tending towards stability, but retains a high degree of unpredictability in the case of random shocks due to underlying market dynamics.