In what follows, I will make liberal use of the notations and terminology from [*ruin theory*](https://en.wikipedia.org/wiki/Ruin_theory), just because I think it makes matters more intuitive. However, the problem I'm posing does not depend on its interpretation within ruin theory. Indeed, the actual original [problem](https://stats.stackexchange.com/questions/536224/the-meaning-of-an-analytical-result-concerning-the-formally-nonexistent-mean-o) that motivated all this had nothing to do with ruin theory.
## The setting
Consider the following discrete random process.
Let $n=1,\,2,\,3,\,\ldots$ (intuitively, you may think of these as dicrete points of time). Define
$$\hspace{8em}R_{n}=u+c\,n-\sum_{j=1}^{n}V_{j}\,,\hspace{8em}(1)$$
where
1. The $V_{j}$'s are independent identically distributed random variables (i.i.d. RVs) whose common probability distribution is such that, with a probability close to $1$, it returns a positive value $v_{\text{D}}$; however, on rare occasions, it instead returns a value greater than some positive $v_{\text{P}}$ from a **Pareto distribution that has an infinite mean.**
In particular, the probability density function is given by
$$\hspace{4em}
f(v)=(1-q)\,\delta(v-v_{\text{D}})+q\,\theta(v-v_{\text{P}})\,\,\frac{({\sqrt{v_{\text{P}}}}/{2})}{v^{3/2}}\,,\hspace{4em}(2)
$$
where $\delta$ is the Dirac delta function, $\theta$ is the Heaviside step function, $v_{\text{D}},\,v_{\text{P}}>0$, and $0<q\ll 1$.
If you find it helpful, you may assume that $v_{\text{D}}$ ('D' for 'deterministic' or 'degenerate') and $v_{\text{P}}$ ('P for Pareto) have some particular values. For example, you might even assume that $v_{\text{D}}=v_{\text{P}}=1$.
2. Note that the $V_{j}$'s are 'arriving' at regularly spaced intervals. This is different (and simpler) than in the standard Cramér–Lundberg model, where their arrival times instead come from a Poisson process (see e.g. [here](https://www.google.com/books/edition/An_Introduction_to_Heavy_Tailed_and_Sube/BSY_AAAAQBAJ?hl=en&gbpv=1&dq=%22In+context+of+the+collective+theory+of+risk,+we+consider+the+classical%22&pg=PA130&printsec=frontcover)).
3. $u\geqslant 0$ is a fixed (i.e. independent of $n$) real number. I suspect that its value won't matter much for what interests me, and that it can be set to zero. I'm including it because people familiar with ruin theory will expect it.
4. $c=(\eta+1)\mu_{V}$.
5. $\mu_{V}>0$ is a fixed real number.
6. $\eta>0$ is a fixed real number.
The meaning of $\eta$ will become clear when I state my conjecture that is the actual subject of this post, below. Basically, it's a 'tolerance level' of sorts.
If the $V_{j}$'s had a finite mean, i.e. if the integral $\int_{0}^{\infty}x\,f(x)\,dx$ existed, then $\mu_{V}$ would be set to that mean. Our main difficulty is that the mean is infinite, so it is not clear to what value $\mu_{V}$ should be set.
In the context of ruin theory,
1. $R_{n}$ is the reserve of the insurance company at time $n$ ('ruin' occurs if $R_{n}$ turns negative);
2. $u=R_{0}$ is the initial reserve;
3. $c>0$ is the rate of inflow of premium;
4. the $V_{j}$'s are insurance claim sizes; and
5. $\eta$ is the *safety loading* (see e.g. [here](https://www.google.com/books/edition/Risk_and_Insurance/w63dDwAAQBAJ?hl=en&gbpv=1&dq=%22and+for+this+reason+denoted+the+safety+loading%22&pg=PA86&printsec=frontcover));
<sup>In the context of insurance, provided the ruin probability isn't 1, $\eta=0$ leads to premium rates ($c$) that are break-even for the insurer. A nonzero $\eta$ leads to net profit.</sup>
We now introduce the notion of ***ruin time*** $\boldsymbol{n}_{\textbf{R}}$ (which in our case is a positive integer): for any given realization of the process $R_{n}$, $n_{\text{R}}$ is the lowest $n>0$ such that $R_{n}<0$.
In ruin theory, the basic question is what is the probability that ruin will occur in infinite time for a given $c$ and $u$. I don't have a proof, but I am fairly certain that this probability is 1 for the process in Eq. (1). I'm basing this on the fact that Kortschak, Loisel, and Ribereau ('Ruin Problems with Worsening Risks or with Infinite Mean Claims', Stoch. Model. **31,** 119–152 (2015), [here](https://www.tandfonline.com/doi/abs/10.1080/15326349.2015.973721), see p. 120) say that in an actual risk process (meaning, a process in which the timing of claims comes from a Poisson process) in which the claim sizes are distributed according to a *pure* Pareto distribution with infinite mean, the infinite-time ruin probability is always 1.
## A vague hypothesis
The intuition about random processes with heavy tails (like Pareto) is that they proceed 'in a normal way' until a single catastrophic jump 'ruins' everything (see e.g. [here](https://www.google.com/books/edition/Risk_and_Insurance/w63dDwAAQBAJ?hl=en&gbpv=1&dq=%22With+heavy+tails,+the+intuition+is+that+of+one+big+jump.%22&pg=PA98&printsec=frontcover)).
My first goal is to formulate a conjecture that will make precise the following vague hypothesis: say we run the process in Eq. (1) and keep track of $\frac{1}{n}\sum_{j=1}^{n}v_{j}$, where the $v_{j}$'s are the actual values of the $V_{j}$'s in that particular run. I want to say something like: as we keep running the process, the average $\frac{1}{n}\sum_{j=1}^{n}v_{j}$ will be converging to a well-defined number $\mu_{V}$—until a catastrophic jump happens. I would like to know (a) is this actually true, (b) if it is, what is the value of $\mu_{V}$, and (c) is there a good estimate (mean? median?) of the ruin time $n_{\text{R}}$.
One reason why I'm invoking ruin theory is that it helps to characterize what is meant by a 'catastrophic jump': it is a jump that results in ruin.
## A more precise conjecture
Here is one attempt at turning the foregoing vague hypothesis into a (more) precise conjecture.
Intuitively, we need $n$ to be large enough so that the 'non-catastrophic' fluctuations in $V_{j}$'s get averaged out. But we also need to condition this on the assumption that the first catastrophic jump hasn't happened yet. One way to make all that precise might be this:
>Set $u=0$. There exists a $\mu_{V}>0$ (a function of $q$, $v_{\text{D}}$ and $v_{\text{P}}$) such that the following holds.
>
>Pick an $\eta>0$, no matter how small, and an $\epsilon>0$, no matter how small. Run the process in Eq. (1) (infinitely) many times. With probability 1, each run will end in ruin, so each run will have some ruin time $n_{\text{R}}$.
>
>Here is the main part of the conjecture: there exists a sufficiently large $n^{*}>0$ such that if we consider only the runs where $n_{\text{R}}> n^{*}$, then, for this group of runs, the following is true: for every $n$ such that $n^{*}\leqslant n < n_{\text{R}}$, the probability that $\left|\frac{1}{n}\sum_{j=1}^{n}V_{j}-\mu_{V}\right|<\eta\,\mu_{V}$ is at least $1-\epsilon$.
The formulation of this conjecture is loosely inspired by Theorem 10.7.2 [here](https://www.google.com/books/edition/Stochastic_Networks/uwHrBwAAQBAJ?hl=en&gbpv=1&dq=%22Next+consider+the+problem+of+formulating+and+showing+that+the+random+walk+behaves+in+its+typical+way+except+for+the+big+jump.%22&pg=PA208&printsec=frontcover) (and, if true, would correspond to some sort of a baby version of that theorem).
## My questions
1. Is my conjecture a good way to make precise the 'vague hypothesis'? Is there a better way?
2. Is the conjecture true? How would one prove it? It seems like the paper by Kortschak, Loisel, and Ribereau I referenced above would be a good start, but they are only considering the ruin probabilities (which they make finite by making $c$ grow with time).
3. If the conjecture is true, what is the value of $\mu_{V}$? Is there an analytic expression? If not, can we at least say how it compares to $v_{\text{D}}$ and $v_{\text{P}}$?
4. How can one characterize the magnitude of ruin time $n_{\text{R}}$? For example, is there such a thing as the mean value of it? Note that this question is independent of whether my conjecture is true.
(This question is an attempt to simplify [this one](https://math.stackexchange.com/questions/4222596/discrete-random-walk-risk-process-characterized-by-a-step-distribution-with-a-he).)