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My question is related to the renewal process, as defined in this document.

Renewal process is an arrival process in which the interarrival intervals are positive, independent and identically distributed random variables denoted $\{X_n; n \ge 1\}$.

$\{N(t); t\ge 0\}$ is a renewal process with inter-renewal random variables $\{X_n; n \ge 1\}$. $N(t)$ is the number of arrivals to a system in the interval $(0,t]$.

See page 3 for lemma 3.1. The first sentence of the proof says:

Note that for each sample point $\omega \in \Omega$, $N(t,\omega)$ is a nondecreasing function of $t$ and thus either has a finite limit or an infinite limit.

What is the sample space $\Omega$ of a renewal process?

My guess: Let's denote the observed value of r.v. $X_i$ by $x_i$.

What is a random variable? It's a function that assigns a value to every outcome from the sample space $\Omega$. If we have the sequence of random variables (that sequence can be infinite), $\{X_n\}$, then we need to define what $\Omega$ is.

I'd say $\Omega$ has to be the collection of all possible infinite sequences of the form $\{x_1, x_2, x_3, ...\}$. Then essentially the random variable $X_i$ is equal to the value on the $i$th position of particular outcome from the sample space $\Omega$. A single outcome is an infinite sequence, because the renewal process consists of an infinite sequence of random variables $X_n$.

Is there anything that should be added or corrected in my definition of the sample space here?

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  • $\begingroup$ I like to think of the points in the sample space as the set of all 'possible continuations' of the actual experiment (or possible worlds). $\endgroup$ – Berci Dec 21 '15 at 11:36
  • $\begingroup$ @Berci the renewal process is just an experiment, just like any other, right? And it has to conform to the definition of probability space. So if you were to define a single outcome of the experiment, would my definition make any sense? $\endgroup$ – user4205580 Dec 21 '15 at 11:40
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Your definition is similar to the usual one, except that you don't want to have separate points $\omega\in\Omega$ for the finite sequences. Just treat them as prefixes of infinite sequences. The $X_i=X_i(\omega)$ you mention are usually called the coordinate functions. So the probability that the first arrival is less than 3 is the measure on $\Omega$ of $\{\omega:X_1(\omega)<3\}$. The measure on $\Omega$ is determined by the distribution of the $X_i$. You might look up Kolmogorov's extension theorem for details. The process $N(t,\omega)$, although it has a continuous time parameter, is determined by the $X_i$, so you can define it on the same probability space, i.e., $N(t,\omega)$ is a random variable on the sample space of sequences, just like the coordinate functions. An event about $N(t)$ is an event about the $X_i$.

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  • $\begingroup$ Regarding the first sentence - is renewal process always infinite, meaning the outcome is an infinite sequence of arrival times? In this case we can forget about finite sequences completely. On the other hand, if it can be finite, then we need to distinguish the outcomes that have the same prefix, but the first one is finite, and the second one is infinite. The idea of sample space is that given the experiment result I can point at one (exactly one) of the element from sample space, so I can't have two elements in sample space that correspond to a single outcome. $\endgroup$ – user4205580 Dec 21 '15 at 17:41
  • $\begingroup$ Do you mean, does the model for a renewal process constrain the # of arrivals to be finite, say 10? I don't think that would be called a renewal process. If you mean, could there be just 10 arrivals and none further, well that would occur with probability 0 assuming $P(X_i=\infty)=0$. $\endgroup$ – snarfblaat Dec 21 '15 at 19:03
  • $\begingroup$ I'm asking whether the sequence of random variables $X_n$ representing the interarrival times is always infinite. My guess is 'yes'. $\endgroup$ – user4205580 Dec 21 '15 at 19:12
  • $\begingroup$ But it isn't clear to me whether you're asking a question about the physical process (i.e., could arrivals stop/there is an infinitely long interrarival time) or about how we model the process (the probability space). As to how we model the process, we can still use infinite sequences, even if some element of that sequence assumes the value $\infty$. $\endgroup$ – snarfblaat Dec 22 '15 at 19:10
  • $\begingroup$ I've updated my description of sample space in the question. Yes, I mean the model. $\endgroup$ – user4205580 Dec 22 '15 at 19:34

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