We study how quantization, occurring when a continuously varying process is approximated by or observed on a grid of discrete values, changes the properties of a Gaussian long-memory process. By computing the asymptotic behavior of the autocovariance and of the spectral density , we find that the quantized process has the same Hurst exponent of the original process. We show that the log-periodogram regression and the Detrended Fluctuation Analysis (DFA) are severely negatively biased estimators of the Hurst exponent for quantized processes . We compute the asymptotics of the DFA for a generic long-memory process and we study them for quantized processes.
We study how the round-off error changes the statistical properties of a Gaussian long memory process. We show that the autocovariance and the spectral density of the discretized process (i.e. the process with round-off error) are asymptotically rescaled by a factor smaller than one, and we compute exactly this scaling factor. Consequently, we find that the discretized process is also long memory with the same Hurst exponent as the original process. We consider the properties of two estimators of the Hurst exponent, namely the log-periodogram regression and the Detrended Fluctuation Analysis (DFA) . By using numerical simulations and analytical considerations we show that the estimators of the Hurst exponent of the discretized process are severely negatively biased . We compute the asymptotic properties of the DFA for a generic long memory process and we apply the result to discretized processes.
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quantization, occurring when a continuously varying process is approximated by or observed on a grid of discrete values, changes the
the round-off error changes the statistical
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long-memory process. By computing the asymptotic behavior of the autocovariance and of
long memory process. We show that the autocovariance and
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, we
of the discretized process (i.e. the process with round-off error) are asymptotically rescaled by a factor smaller than one, and we compute exactly this scaling factor. Consequently, we
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quantized process has
discretized process is also long memory with
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of
as
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show that the
consider the properties of two estimators of the Hurst exponent, namely the
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are severely negatively biased
. By using numerical simulations and analytical considerations we show that the
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for quantized processes
of the discretized process are severely negatively biased
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asymptotics
asymptotic properties
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long-memory
long memory
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study them for quantized
apply the result to discretized