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I'll do my best to explain what I'm looking for with a discrete example.

Suppose that I have an event that happens everyday and I measure it.

If this event were to happen exactly one time a day, my measurements for n days would look like: [1, 1, 1, ... n times].

For this particular case, an estimate on how many events I would find after n days, could be given by a function: $f(days) = days * 1$

But, if this event could happen many times a day (randomly), a measurement would look instead like: [2, 1, 0, 2, ... n times]

It's easy to construct a function: $f(days) = ...$ that still estimates the amount of events that happened after n days. But I would like to do so in a way that guarantees that $f(day)$ is always smaller (or equal) than $f(day+1)$, $f(day+2)$, etc...

Is there a mathematical construct that allows this?

The naive solution to my problem would be to simulate a stochastic process that generates events every day and accumulate the result.

But I'm looking for a function that can be evaluated quickly, only takes the number of days and doesn't involve recursion. Something that guarantees me that as the number of days is increased, the number of events do so as well, but never at the same rate.

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    Any such function is monotone. It sounds like your constraint is that the function doesn't increase linearly. – 2017-01-29
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    You're right. Non-linear but having the property I described. Any suggestions? – 2017-01-29
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    There are a ton of functions like this. $f(x) = x^2,x^3,$ etc. In real life $f$ would probably be something involving the Poisson distribution. – 2017-01-29
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    Sorry man, but those functions are not random... like at all... And Poisson well, it's just not what I'm looking for. – 2017-01-29
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    OK. There is no such thing as a "random" function. I have no idea what you are asking for. – 2017-01-29

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