Renewal – TJ Personal Website

As promised, I finally got around to reading the relevant two chapters of Adventures in Stochastic Processes. Bits and pieces of which I found informative and interesting, but most of which unfortunately went over my head. For one thing, the book is designed for a full semester graduate course in Stochastic Processes, so trying to read and comprehend in a couple of days simply isn’t the way this text was designed to be read. What I would have to do in order to really appreciate the material is take diligent notes like I have posted on the contacts page for material we learned in Probability Theory and in Inference I (which is still incomplete). And this is something that I wouldn’t mind doing, except that I successfully got the TA assignment for Asymptotics and I really would be upset with myself if I didn’t devote all of my remaining break time to preparing myself for that. So instead what I’m going to do is make some brief notes about some the stuff I read in the chapters here, and then I’m going to make as many good notes as I can for Asymptotics (I’m very excited to crack open the Van der Vaart textbook I got for Christmas), and then eventually maybe I’ll come back and take some real notes for Stochastic Process stuff here.

But first, why did it take me so long to get around to this? I actually was having a bit of a ‘renewal’ process myself, taking some time of the break off of anything related to Stats and doing something that I love doing and haven’t had enough time to do lately: reading. I also spent a good deal of time with family and friends. In the days between Christmas and now I read “In Cold Blood”, “The Kite Runner”, and “Cat’s Cradle”, watched with family (or my girlfriend) the movies Annie Hall, Manchester by the Sea, Predestination, LOL, 2001: A Space Odyssey, and also a lot of pop-culture jeopardy with Colin Jost. To top it off, my brother and I beat Halo: Reach on legendary difficulty. All in all, I had an immensely gratifying Christmas & New Years, and now I go back to the grindstone.

First, I’ll talk through chapter 3, then chapter 4. In retrospect only Chapter 4 will likely be helpful for Survival Analysis, but nevertheless I read chapter 3 so I have to talk about it a bit. Chapter 3 is about Renewal Processes. From the book “Renewal Processes Model occurrences of event happening at random times where the times between events can be approximated by independent, identically distributed random variables.” From the last blog post we know that if these random times between events are iid exponential, then we have a poisson process. A renewal process sort of “restarts” itself every time it has a “renewal” which is an event, and there can be multiple types of randomness embedded in the renewal process. One example that was referenced a few times throughout the chapter was that of insurance claims: the claims appear at random times, e.g. in a poisson process, but the thing of interest is not the claims but the amount of money the insurance company has (e.g. how likely it is to go bankrupt). As an aside, for those who might be thinking “insurance companies never go bankrupt”, with poor modeling insurance companies and in fact entire industries can go bankrupt. When I worked at Excellus BCBS, which is a non-profit, one of the member companies (“Excellus” is just one arm of what is known as the “Lifetime Healthcare Companies”) called MedAmerica was ironically inaugurated as a way to discount the non-profit segments of the company by bringing in money with a for-profit segment. The for-profit MedAmerica Long Term Care Insurance policies that were based on assumptions about interest rates in the 90s staying forever as if it were the 90s and assumptions about the cost of Long Term care remaining relatively fixed (and unrealistic assumptions about the number of people that would drop coverage after they bought it) that were simply not tenable. MedAmerica ended up hemorrhaging cash and now no longer sells policies, tries to get the government to allow rate increases every year, and ended up costing the non-profit segments a lot more than it ever gained. However, at least the Lifetime Healthcare Companies are able to honor all of the policies they’ve sold. Many companies that sold that type of Long Term Care insurance simply went bankrupt, more or less robbing the policy-holders of the coverage that they had invested in for later in their lives. Anyway, this insurance model they talked about in the textbook is more of an example and doesn’t really have anything to do with long term care, the story just popped into my head so I wrote it.

Anyway, the company accumulates money at a constant rate throughout time (premiums), and when a claim comes in (e.g. in a poisson process) the company then loses some random amount of money. This is a renewal process. Immediately after the claim, it’s as if the company is ‘starting again’ with however much money is currently in the bank after that claim. Another example is a process of up and down times of a machine. A machine operates and goes down at random times controlled by one renewal process, and each time it goes down the time until the machine is back up again is controlled by a different renewal process.

Often, the first event is more likely to happen sooner than subsequent events. For example, let’s say you arrive at a bus stop, and busses come every ten minutes with a standard deviation of one minute (not much variation between interval lengths). If you arrive at a random time you’ll probably only have to wait about five minutes for the first bus, and then you’ll have to wait about ten minutes for each subsequent bus. This is called a delay, and as a result the first event has a distribution that can differ from the distribution of all subsequent events. If the renewal process is Poisson of course this is not an issue (due to memorylessness).

The convolution of a function g(t) defined in R+ locally bounded and a distribution F is defined as $F * g(t) \equiv \int_0^t g(t - x) F(dx)$ for $t > 0$ , noting that this integration includes endpoints. Essentially, this can be thought of as the weighted area under the curve where the weighting is done by the distribution, and the functions you’re weighting are shifted versions of the function g. For example, the convolution of $g(t) = \sin(t)$ and the bernoulli distribution is, because we only have probability mass at x = 0 or x = 1, just $(1-p)sin(t) + p \sin(t -1)$ , so if p is 0.25 you’re weighted more towards sin(t) and if p is 0.25 you’re weighted more towards $sin(t-1)$ . In the little graph on desmos below you can see how this convolution works, at any point on the red you are just 0.75 times the value of the blue plus 0.25 times the value of the green.

The LaPlace transform of a non negative random variable X is defined as $\hat{F}(\lambda) = E[e^{-\lambda x}]$ for $\lambda > 0$ . Note that the way the transform is defined $\hat{F}(\lambda) < \infty$ for all $\lambda$ , like Moment Generating Functions each distribution has a distinct LaPlace transform.

The counting function of a process is the number of renewals in [0, t], N(t), and of course is a random variable. For a poisson process N(t) is poisson with rate proportional to t. The Expected Value of N(t), $U(t) = E[N(t)]$ is called the renewal function, and since it is an expectation taken across a random variable that incorprates t, it is simply a function of t (and is not random). For example, in the poisson process it is simply $\lambda t$ . In the delayed case, the first waiting time having distribution G and all others having distribution F, the renewal function is the convolution $V(t) = G + G * U(t)$ . If we are considering the first event as the ‘zero-th’ (as the book does) then it’s just $V(t) = G * U(t)$ .

One of the main asymptotic results of renewal theory is Blackwell’s Theorem, which says that under certain conditions $V(t, t + a] \rightarrow a/u$ as $t \rightarrow \infty$ , which basically says that the number of events goes to the rate of events, if the interarrival distribution has a mean of 5 (on average there are 5 minutes between events), then the expected number of events in an interval of length a approaches a/5, or an event every “5” time units waited.

A delayed renewal process is called stationary if the interarrival times are iid with a common distribution $F$ and finite mean, and a delay distribution $F_0$ again with finite mean.

Chapter four focuses on point processes, which are directly connected to renewal processes in that point processes count the number of renewal epochs. If E is a subset of the euclidean space, say $\mathbb{R}^1 +$ , and points are distributed randomly on this interval (e.g. a poisson process) and the count of the number of points in any interval A is N(A) (that is N is the counting measure), then N is a point process and $X_n$ are the points.

A process for which N(A) is a poisson random variable and for which all disjoint subsets $A_1, ..., A_n$ of E have $N(A_1), ..., N(A_n)$ independent is called a poisson process. The poisson process can be transformed in various ways. For example, let’s say a poisson process on the real line is mapped to its square, that is a point located at 0.5 is now located at 0.25, a point located at 1.5 is now located at 2.25, etcetera. The measure in the original from 0 to sqrt(t) is equivalent to measuring from 0 to t in the new. A poisson process is thinned if every time a jump would occur, it only occurs with probability p. A thinned poisson process is also a poisson process with a longer average interarrival time.

Hopefully I have enough context now that I won’t be entirely lost when I take survival analysis in the spring. Now, I return to my main task: finishing the notes for Inference 1, and writing self study notes for all of inference 2 so I can TA it in the spring without making too much of a fool out of myself. I’m actually very excited about this upcoming semester.

Happy 2025!

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