 Independent-record averaging versus high detector bandwidth

Independent-record averaging and increasing detector bandwidth change a measurement in different ways. Let $Y_1,\dots ,Y_N$ be independent measurements of zero-mean noise, made with the same fixed filter, and suppose each has finite variance ${\sigma }_B^2$. Their average satisfiesThe vanishing mean is an ensemble statement; the finite-record noise fluctuations only decrease as the number of independent records grows. For correlated records the variance is instead $N^{-2}{\sum }_{i,j}\mathrm{Cov}(Y_i,Y_j)$, so merely increasing the number of samples need not give the same reduction.

Bandwidth controls which fluctuations enter a record. With two-sided angular-frequency power spectral density $S(\omega )$ and an ideal low-pass filter retaining $|\omega |\le \Omega$, the filtered variance isFor flat noise spectrum $S(\omega )=S_0>0$ in that range, this is $S_0\Omega /\pi$, and averaging $N$ independent records gives $S_0\Omega /(\pi N)$. Increasing bandwidth exposes more of the physical fluctuations; independent averaging reduces their variance. Sampling faster from one correlated trajectory is not automatically equivalent to obtaining independent records.