Does the concept of "dynamic average" makes any sense? While making an excel table about how many times an event happens per day I thought that it could be interesting to see what is the growth rate of those events.
If in $2$ days the event happens two times on the first day and three times on the second, I have a an average of $X_1 + X_2/2=2.5$ per day.
So I had the idea to calculate the average of the first $n$ days as $\frac {\sum^{n}_{i=0}X_i}{n}$, where $X_i$ is the number of events that occurred on the $i$th day.
If $f:\mathbb N \rightarrow \mathbb N$ describes the number of events, the "dynamic average" can tell us if the average is growing or not:
$$\mathfrak M(f)(n)=\frac {\sum^{n}_{i=0}f(i)}{n}$$
$\mathfrak M(f)(n)$ should tell us how many events/day we have the $n$th day.
I'm not sure if this has a name (probably yes), but I was wondering what happens if we want to see the "dynamic average" of a an event/quantity that doesn't happen once a day but always at any time.
I understand that explained in this way the concept makes little sense and is inaccurate, but let's see it as a function on the real, where the real axis is a time line. $f(x_0)$ tells us what happens at the moment $x_0$.
I tried the following formula. 
$$\mathfrak M^*(f)(x)=\lim_{n\rightarrow +\infty}\frac {\sum^{n}_{i=0}f(i\frac x n)}{n}$$
The idea is to take alot of little strips of decreasing width in order to have infinite cases in the time interval $[0,x]$, and with infinite events we take the average of all the events that occour before the moment $x$.

Is this a good way to catch the concept of "dynamic average"?
How the problem was tackled in statistics? What is the right way to
catch this concept?

I've plotted some graph and I see that $\mathfrak M(f)(n)$ is discontinuous, while $\mathfrak M^*(f)(n)$ is continuous. 

What is the relation between $\mathfrak M$ and $\mathfrak M^*$ and what is the big picture?


I'm really sorry for my horrible english. I used google translator massively but I still think I've made some big mistake somewhere, sorry again.
 A: The description of your discrete case suggests you are trying to look at something like a moving average,, except in your case the start of the window is fixed (e.g., $i=1$), while the end of the window $n$ is extended. Yes, it makes sense to look at this. Indeed, many financial analysts use it with stock market data etc.
You then mention a continuous case. In statistics, you would probably represent $f(x)$ as the density of a stochastic arrival process. One such process is the Poisson process. The density describes how many events will take place in a given time period $[t,t']$. However, such a density is usually used to calculate the expected number of events, while you apparently already know the realizations of this process, i.e., you have empirical data.
I am not sure what your goal is, but if you want to calculate the moving average of that data not per day, but per unit of time as apparently in your $M^*$ version, then why don't you define $f(x):\mathbb{R}_+\to \mathbb{N}$ as the number of events that occurs up to time $x$? Then
$$M^*=\frac{f(x)}{x}.$$
That would be the straightforward extension of your discrete version of "dynamic average", and it would be discontinuous just like the discrete version: every time an event occurs, the average jumps up a bit.
A: Below link to the paper Dynamic programming: Arithmetic, Geometric, Harmonic, and Power Means.
https://www.researchgate.net/publication/363488132_Dynamic_Programming_Arithmetic_Geometric_Harmonic_and_Power_Means
https://www.academia.edu/resource/work/86545241
https://github.com/calebnwokocha/manuscripts/blob/main/Dynamic%20Programming%20-%20Arithmetic%2C%20Geometric%2C%20Harmonic%2C%20and%20Power%20Means.pdf
