>nad, I thought you were concerned about the sign of tau, so I probably just confused you.

No. I interpreted this as a typo, i.e. instead of $$C_{i,j}^t(-\tau) $$ as David mentioned above they probably meant:

$$ C_{i,j}^{t-\tau}(-\tau)$$

I am concerned about their interpretation of correlation, it might not be what they originally wanted and not be what they think it represents.

No. I interpreted this as a typo, i.e. instead of $$C_{i,j}^t(-\tau) $$ as David mentioned above they probably meant:

$$ C_{i,j}^{t-\tau}(-\tau)$$

I am concerned about their interpretation of correlation, it might not be what they originally wanted and not be what they think it represents.