Discussed in this paper are statistical properties of correlation functions calculated from data of finite length.
First, a multi-dimensional characteristic function of sample covariances of signals, which are generalized expressions of the calculated correlation functions, is expressed as a function of the signal spectral matrix, assuming that the signals are ergodic and gaussian. Covariance of any two correlation functions is calculated from this characteristic function.
Secondly, a process to measure the equivalent gain of a system from its input and output data is analysed. If the signal bandwidth is much smaller than that of the system, the variance of the measured gain normalized by the mean squared is smaller than 2/n times (noise to signal ratio), while normalized variance of input-output cross-correlation function is larger than 2/n, where n is the data length times the input bandwidth.
Finally, a process to calculate the frequency response in real time is analysed. The probability density function is approximately calculated. The normalized variance of the frequency response is found to be smaller than 1/n times (noise to signal ratio), where n is the data length times the equivalent bandwidth of the resonant filters used for extraction of each frequency component from the input and the output.

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