This chapter discusses theoretical autocovariance, autocorrelation functions of autoregressive models of orders 1 and 2, and autocorrelation function-derived timescale. The autocorrelation function of a scalar time series is a prime tool for estimating the characteristic timescale separating successive independent realizations in the time series. Any process other than completely random noise has some serial correlations. Even variables as volatile and nondeterministic as measures of stock market performance, when valuated close enough, are unlikely to vary appreciably. Thus, a time series of the index at 1-second intervals likely contains significant redundancy; it can be almost as representative and contain almost as much information if degraded to valuation intervals of T > 1 second. Identifying an acceptable characteristic timescale T separating successive independent realization in the time series that balances the need to retain maximum information while minimizing storage and transmission burdens is a key role of the autocorrelation function.
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