Physics of Risk

Approximating characteristic function of bounded Pareto distribution with α=1

Aleksejus Kononovicius — Tue, 08 Sep 2026 08:00:00 +0300

During the summer, I received an email from a researcher at university abroad inquiring about how we (myself and my senior colleague, prof. B. Kaulakys) derived the approximation for the characteristic function of bounded Pareto distribution (for the particular case with \( \alpha = 1 \)). The approximation was given in [1], but derivation turned out to be somewhat more involved than I remembered. Originally, I derived it with the help of Wolfram Engine, while my colleague consulted various mathematical formula compendiums. Unfortunately, I was unable to find any notes containing the derivation. So I have attempted to reconstruct a combined approach from scratch. You'll find the derivation below.

We are back for a new academic year (2026)!

Aleksejus Kononovicius — Tue, 01 Sep 2026 08:00:00 +0300

So, we are back for a new academic year! Over the summer, our group continued reading books on statistics. Our summer readings, both old and new, has inspired a few posts that will be published on the blog soon. We hope you will find them enlightening!

Oh, and by the way, this September is our 21st September since we started Physics of Risk blog (if we include the 2006–2010 period when the posts were written in Lithuanian). 20 years have gone by!

Road trip inspired by a calendar

Aleksejus Kononovicius — Tue, 23 Jun 2026 08:00:00 +0300

Do you have plans for summer holidays? If not, here is a quick idea presented to you by Woolly Benguin at MathsJam 2023.

Beta prime distribution from Gamma-distributed random values

Aleksejus Kononovicius — Tue, 16 Jun 2026 08:00:00 +0300

As we have already discussed, beta prime distribution arises from a nonlinear transformation of the voter model. Furthermore, recently we have been relying a lot on the said transformation [1, 2, 3]. In those papers, we have been using some results derived for the CIR process as well. Thus, another interesting thing, which my colleague Rytis Kazakevičius has noted, was that the beta prime distribution can be obtained from the ratio of two independent Gamma-distributed random values. Why it is interesting? The stationary distribution of the CIR process is the Gamma distribution!

Beta prime distribution

Aleksejus Kononovicius — Tue, 02 Jun 2026 08:00:00 +0300

Have you heard of the beta prime distribution before? Until recently, I hadn't either. My colleague, Rytis Kazakevičius, recently surprised me by pointing out that the distribution we have been repeatedly encountering in nonlinear transformations of the noisy voter model has actually a proper name. It is known as the beta prime distribution. And our history with this distribution, goes back much further.

MinutePhysics: Problem with Newcomb's paradox

Aleksejus Kononovicius — Tue, 19 May 2026 08:00:00 +0300

Recently, Veritasium has posted a video on Newcomb's paradox. This paradox is based on a decision-making problem. Namely, you are presented with two boxes. Box A is transparent and contains a small amount of money, while Box B is opaque. Box B might contain a large amount of money or be empty. Its contents are decided ahead of time by a machine, which has perfect prediction record. If the machine predicts that you will take only Box B, it will put large amount of money inside it. Otherwise, it will keep Box B empty. So, will you take only Box B (and believe in predictor accuracy) or will you take both boxes (and believe in game theory)? For more details on the paradox and its solutions watch the video below.

After watching the video above I wasn't particularly fond of the paradox. Mostly because the paradox seemed to be artificially contrived. After watching minute physics (previously known as One Minute Physics) video I still feel the same way, but surprisingly this paradox prompts us to consider the nature of our own reality. I encourage you to watch it as well.

I especially liked the part about random 5 year old falsely achieving incredible accuracy.

The defining property of stable distributions

Aleksejus Kononovicius — Tue, 12 May 2026 08:00:00 +0300

Our group, along with a few students, has been reading statistics handbook and refreshing our understanding of the basic statistics. I was given to cover a chapter about the central limit theorem, which reminded me that I had already given a similar presentation while being PhD student myself. While diving into the topic, I have noticed a couple things, which are usually glanced over in a typical statistics handbook. In the final post of this series, let me put an emphasis on the defining property of any stable distribution.

Summation of infinitely divisible random variates

Aleksejus Kononovicius — Tue, 28 Apr 2026 08:00:00 +0300

Our group, along with a few students, has been reading statistics handbook and refreshing our understanding of the basic statistics. Some time ago, I was given to cover a chapter about the central limit theorem, which reminded me that I had already given a similar presentation while being PhD student myself. While diving into the topic, I have noticed a couple things, which are usually glanced over in a typical statistics handbook. Let me share them with you.

This time we explore infinite divisibility. Our previous mathematical explorations of the stable distributions topic have relied on this property, because it simplifies many of the analytical derivations. But there are distributions, which are infinitely divisible but not stable. This time let us take a look at sums of Gamma distributed random variates.

A. Chadha: Cauchy 101

Aleksejus Kononovicius — Tue, 21 Apr 2026 08:00:00 +0300

Over the last few posts we have taken a look at the Cauchy distribution, which apparently has undefined mean and variance. You will find a useful review and some novel personal insights from an enthusiastic student Ananya Chadha in the video below.

MSCA hosting invitation (2026)

Aleksejus Kononovicius — Thu, 16 Apr 2026 08:00:00 +0300

If you are interested in complex systems, physical modeling of opinion dynamics or financial markets, then you might be interested in my invitation to host MSCA postdoctoral fellows.

More information on the Euraxess website »