Sigma notation with a variable in the lower limit

Question

I have witnessed the following mathematical notation in various scholarly journals:

$$ \sum_{t}^{} $$

where the lower limit only contains the variable $t$. In these circumstances, the person is usually multiplying one variable $x$ by another variable $y_{t}$ observed across time, where $t = 0, 1, 2, ..., T$. For example,

$$ \sum_{t}^{} (x*y_{t}) $$

where the limits are not explicitly indicated.

Question: Does a single variable in the lower limit denote the entire interval? Or, does this notation implicitly generalize to any possible interval?

Answer 1

In general, the notation

$$\sum_t$$

means that you should sum over all possible values of $t$. If you have $n$ data points labeled $y_1$, $y_2$, $\ldots$, $y_n$ then you can read this notation as

$$\sum_t\equiv \sum_{t=1}^n$$