# Covariance between normalised correlation functions

If I have a set of correlated random variables $\mathbf{X}=\{X_1,\dots,X_n\}$ that have been sampled $N$ times, I can calculate the correlation function for pairs of variables as $$\mathrm{Corr}(i,j) = \frac{1}{N}\sum_k x_{i,k}x_{j,k} \equiv C_{ij}$$
where $x_{i,k}$ is the $k^{\mathrm{th}}$ realisation of the random variable $X_i$. The covariance between two correlation elements can be calculated as $$\mathrm{Cov}(C_{ij},C_{lm}) = \mathrm{Cov}( \frac{1}{N}\sum_k x_{i,k}x_{j,k},\frac{1}{N}\sum_{k'} x_{l,k'}x_{m,k'} )\\ = \frac{1}{N^2} \sum_{k,k'} \mathrm{Cov}( x_{i,k}x_{j,k}, x_{l,k'}x_{m,k'} ),$$ where the covariance is defined as the expectation value $$\mathrm{Cov}(a,b) = E[(a-E[a])(b-E[b])].$$ If the samples are independent we have $k=k'$ and as long as we know the correlations between all the random variable $X_1\dots X_n$ we can calculate the covariance. However, what about the covariance between elements of the normalised correlation function defined as $$\mathrm{Ncorr}(i,j) = \frac{\mathrm{Corr}(i,j)}{\bar{X}_i \bar{X}_j}= \frac{\frac{1}{N}\sum_k x_{i,k}x_{j,k}}{\frac{1}{N}\sum_p x_{i,p} \frac{1}{N}\sum_t x_{j,t}} \equiv g_{ij}.$$ Is it possible to write $$\mathrm{Cov}(g_{ij},g_{lm}) = \frac{1}{N^2} \mathrm{Cov}\left( \frac{\sum_k x_{i,k}x_{j,k}}{\frac{1}{N}\sum_p x_{i,p} \frac{1}{N}\sum_t x_{j,t}},\frac{\sum_{k'} x_{l,k'}x_{m,k'} }{\frac{1}{N}\sum_{p'} x_{l,p'} \frac{1}{N}\sum_{t'} x_{m,t'}}\right)\\ = \frac{1}{N^2} \frac{1}{\frac{1}{N}\sum_p x_{i,p} \frac{1}{N}\sum_t x_{i,t}} \frac{1 }{\frac{1}{N}\sum_{p'} x_{l,p'} \frac{1}{N}\sum_{t'} x_{m,t'}}\sum_{k,k'} \mathrm{Cov}( x_{i,k}x_{j,k}, x_{l,k'}x_{m,k'} ) \\ = \frac{\mathrm{Cov}(C_{ij},C_{lm})}{\bar{X}_i \bar{X}_j \bar{X}_l \bar{X}_m}$$ or is this incorrect since you are pulling correlated variables outside of the $\mathrm{Cov}$ operator?

• You lost me at the very first line, which is neither the sample correlation nor a covariance. (It is a sample raw second moment.) Are you sure this is the expression you want to analyze? At the second link, since $k$ is a bound variable in the summation, the reference to it is meaningless. The "normalization" in the third step makes no sense because the denominator could be zero. – whuber Oct 16 '14 at 15:18
• Ok then I guess I'm interested in the covariance between "normalised" sample raw second moments. I realise that the normalisation looks odd since, as you point out, the denominator could be zero but in my case they are not. Supposing the function $g_{ij}$ is well defined, calculating the covariance should be perfectly valid. – mrkprc1 Oct 16 '14 at 15:54
• Given, then, that you are using terms like "correlation" in nonstandard ways, your readers cannot assume anything. I must therefore ask how you are conceiving of the "covariance". Is that intended to be an expectation or is it supposed to be some moment of the data? Because you refer to random variables, it would seem the former, but in that case whether your realization has a nonzero denominator is of no help: what matters is whether there is any chance it could have been zero. – whuber Oct 16 '14 at 16:24
• Your right, I'm meaning the expectation value. I'll add that to the question. – mrkprc1 Oct 16 '14 at 16:29