I'm reading a paper which has the following description;
Say we have a time series of correlated sequential observations of the random variable $X$ denoted $\{x_n\}_{n=1}^N$ from a stationary, time reversible stochastic process.
Expectation of $X$ is
$\hat{X} = \dfrac{1}{N}\sum_{n=1}^Nx_n$
and statistical uncertainty is
$\delta^2\hat{X} \equiv \langle(\hat{X}-\langle\hat{X}\rangle)^2\rangle = \langle\hat{X}^2\rangle - \langle \hat{X} \rangle^2$
Which can be written as
$\delta^2 \hat{X} \equiv \dfrac{1}{N^2}\sum_{n,n' = 1}^N \bigg[ \langle x_n x_{n'} \rangle - \langle x_n \rangle \langle x_{n'} \rangle \bigg]$
Focussing on the summation, what is meant by the $[n,n'=1]$ notation? I'm suspect it's shorthand for
$\delta^2 \hat{X} \equiv \dfrac{1}{N^2}\sum_n^N\sum_{n'}^N \bigg[ \langle x_n x_{n'} \rangle - \langle x_n \rangle \langle x_{n'} \rangle \bigg]$
But I don't know why you wouldn't just write that.
EDIT1 Moreover, in this context, it's not clear to me why $\langle x_n \rangle$ and $\langle x_{n'} \rangle$ are not simply both equal to $\hat{X}$, assuming we read $\langle x_n \rangle$ as "the expected value of a member of the set generated by the random variable $X$". Or perhaps we should read $\langle x_n \rangle$ as the series from $n$ to $N$, such that as we increment $n$ the series of values becomes shorter (which I think makes intuitive sense)?
EDIT 2 Sorry - ignore all that, $\langle x_n \rangle$ is explicitly defined as the expected value of $x_n$ over repreated independent trials (i.e. Generating 100 independent series and determinig the average value of the $n$th member).
It's a double summation.
$$\sum_{n=1}^N\sum_{n'=1}^N$$
Of course the given notation is more convenient.