Over-indexing and undefined expressions

Reading the literature, we often encounter sums of the form

\displaystyle\sum_{x_k\in\Phi} f(x_k)

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?

\displaystyle\sum_{x_k\in\Phi} f(k)

To make this concrete, let us focus on this simple form:

\displaystyle\sum_{n_k\in\{1,2,3\}} k^2

What is the result? It is not 14 but undefined, since k is undefined. This problem occurs fairly frequently in the form

\displaystyle\sum_{x_k\in\Phi} h(k)f(x_k)\qquad\text{or}\qquad \sum_{x_k\in\Phi} h_k f(x_k)

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

\displaystyle\sum_{k\in\mathbb{N}} h(k)f(x_k),\quad\text{where }\Phi=\{x_1,x_2,\ldots\}.

Alternatively,

\displaystyle\sum_{x\in\Phi} h_x f(x).

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.

3 thoughts on “Over-indexing and undefined expressions

  1. Good one Martin, what do you think about the following notation:
    $\sum_{i\in\Phi}h_{i}l(r_{i})$ in this case I believe the over-indexing issue can be avoided and also if instead of h we wanted to use ho for instance, i.e. gain between ith node and node placed at the origin it could be explicitly stated.

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    1. I am not a fan of using $i$ to represent a point of a point process; most people use $i$ for integers, so using it for an element of R^d may be confusing. While there is nothing mathematically wrong about using $i$ here, $\sum_{x\in\Phi}h_{x}l(r_{x}$ is better notation. Similarly, most people probably prefer to write (and read) $\int_{x=0}^10 f(x)dx$ for example, rather than $\int_{i=0}^10 f(i)di$.
      Next, I am not sure how to connect $i\in\Phi$ with the “$i$-th node”. The latter would suggest that $i$ is an integer, but if $i\in\Phi$, this is definitely not the case. So that is an inconsistency.

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      1. Thank you, Martin, for the prompt and interesting response. I was just thinking back on the notation used for modeling dyadic fading where notation slightly gets complex and defining using $i$ could simplify life but I agree with your point about this is not a preferable option as connecting $i\in\Phi$ with the “$i$-th node” interpretation might be misleading. Thanks for your valuable feedback.

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