卷积定义中反向索引配对的实用原因与信息捕捉机制问询

I am taking some time to carefully construct my understanding of generating functions as they are extremely interesting to me in terms of properties and uses.

I am making my way to convolutions now and the formula goes:

For sequences a = {$a_i:i\ge0$} and b = {$b_i:i\ge0$} we have that their convolution $c$ is $$c_n = a_0b_n+a_1b_{n-1}+…+a_nb_0$$ $$=\sum_{i=0}^na_ib_{n-i}$$

My question is why we choose to define the convolution by pairing the $(i)th$ moment of $a$ with the $(n-i)th$ moment of $b$ as opposed to equating the indices for both. What information is being captured by having the sequences run in opposite directions? I tend to come at this from relating the information to probability theory but any approach is welcome.

Great question — this reverse-index pairing isn't some arbitrary choice; it's tied directly to the problems convolution was invented to solve, especially when you link it to generating functions and probability (your go-to angle!). Let's break this down:

1. It's the natural result of multiplying generating functions #

First, remember that the generating function for sequence $a$ is $A(x) = \sum_{i=0}^\infty a_i x^i$, and for $b$ it's $B(x) = \sum_{j=0}^\infty b_j x^j$. When you multiply these two:
$$A(x)B(x) = \left(\sum_{i=0}^\infty a_i x^{i\right)\left(\sum_{j=0}}\infty b_j x^j\right) = \sum_{n=0}^\infty \left(\sum_{i=0}^n a_i b_{n-i}\right)x^n$$
The coefficient of $x^n$ is exactly that reverse-paired sum. So convolution is just the operation that corresponds to multiplying generating functions — which is why it's so central to combinatorics and generating function theory. If we used same-index pairing ($\sum a_i b_i$), that would be a dot product, which doesn't have this nice multiplicative link to generating functions.

2. Probability: Capturing sums of independent random variables #

Since you're coming at this from probability, let's lean into that. Suppose you have two independent non-negative integer-valued random variables $X$ and $Y$. The probability that their sum is $n$ is:
$$P(X+Y = n) = \sum_{i=0}^n P(X=i)P(Y = n-i)$$
This is exactly the convolution of the PMFs of $X$ and $Y$. The reverse pairing here makes perfect sense: for the total to be $n$, if $X$ takes value $i$, $Y$ has to take the remaining value $n-i$. We're enumerating all possible pairs of outcomes that add up to $n$ — that's the information being captured here: all combinations of individual outcomes that sum to a total $n$.

If we used same-index pairing ($\sum P(X=i)P(Y=i)$), that would be the probability that $X=Y$, which is a completely different quantity with a totally different use case.

3. Other contexts: Capturing "shifted interaction" #

Outside probability, think about signal processing. When you convolve an input signal with a filter, you're essentially flipping the filter and sliding it over the signal. The reverse index pairing here captures the idea of "how much the filter's $i$-th sample overlaps with the signal's $(n-i)$-th sample at position $n$" — it's measuring the cumulative effect of the filter on the signal over time, accounting for delays.

To sum it up #

The reverse index pairing is all about capturing combinations of elements that add up to a fixed total (whether that's a sum of random variables, a degree in a generating function product, or a time offset in a signal). Same-index pairing is a different operation (dot product) that measures similarity or coincidence, not cumulative additive combinations.

备注：内容来源于stack exchange，提问作者Philo-Sophism