Reading the literature, we often encounter sums of the form
for a point process Φ. This is an instance of what may be called “over-indexing”, since the subscript k is clearly not needed. It makes the notation unnecessarily cumbersome, but there is nothing mathematically wrong with it. If it is tacitly assumed that using a dummy variable of the form xk somehow defines k, problems arise. For instance, what exactly is meant by this formula?
To make this concrete, let us focus on this simple form:
What is the result? It is not 14 but undefined, since k is undefined. This problem occurs fairly frequently in the form
where h(k) is used to denotes a fading random variable indexed by some (undefined) k and f is a path loss function. What is meant is usually
Here the fading random variables are indexed by the points, i.e., they can be interpreted as marks associated with the points. This is the form I personally prefer, for it does not require a particular way of indexing the points and it is more compact.