Better Call SLLN: Black Matter Lives

“The future ain’t what it used to be.” –Yogi Berra

 

Almost surely (“a.s.”), you have heard of, or even watched, the tv-series Breaking Bad (BB). Almost as surely (“in probability”), you know many matters do not always go the way you expect. Laws matter in BB, as they do in quantitative trading, too: with a high probability, you get punished if breaking the law. But not “almost surely” – and this difference matters.

 

Let us try to decipher that. In what follows, we expect a fair bit of knowledge of BB, at least tangential interest in quantitative trading, statistical knowledge, and adherence to “law.” Laws are data-driven: you may be unaware of any laws being broken in your locale until evidence amasses in large numbers. When that happens, “[somebody] better call Saul.”

 

Suppose for a moment you are a lawyer yourself – just for illustration purposes, God Bless – and you have already gained a large number of law cases under your belt. What is then the probability of winning the next case? Well, you may have a certain expectation for that. This expectation is calculated based on your historical track-record: often, the common average.

 

The more cases you tackle, the more precisely this average reflects your “true talent”; your success average gets closer to the “truth,” like it not, revealing your inherent skill. The cases you have tackled do not have to be identical in terms of how hard they are nor do they need to be independent of each other. Like black magic, the Law of Large Numbers (LLN) delivers.

 

Similarly, in quantitative trading, “win-rate” is the number of successful trades divided by all of your trades. When the number of cases becomes large, data turn into “Black Matter.” In BB, black matters in the form of Huell. But there are cases where this “Saul’s Law of Large Numbers” does not apply to, that is, Huell cannot enforce, and we discuss about that next.

 

Not even the Bureau of Land Management (BLM) can manage the unpredictable cases – the infamous Black Swans. It turns out there is indeed a thin red line between Black and White: since Black Swans are by definition unpredictable and may cost a fortune when they occur, dumping all of White’s cash in one place is not the advised “Black Swan defensive strategy.”

The better-known Bureau, the Federal Investigation one, typically handles the Black Swans: To catch the likes of Mr. White, a special file is opened as outliers do not follow the Normal (“Gaussian”) Law. In statistics, the Weak LLN is routinely proven with only Gaussian random variables, but the Black Swans, although uncorrelated, create extremely large fluctuations.

 

Surprisingly, it can be proven that Black Swans do not pose any serious threat to either the Weak (“P”) nor the Strong (“a.s.”) forms of LLN. Pushing the boundary of law, Saul handles the Black Swans uniquely, which on Netflix is his trademark and main value proposition. So, Black Matter is not directly related to Black Swans, but we are getting closer to the truth.

 

Observation frequency matters: As high-frequency traders know, Black Swans tend to occur relatively frequently in intraday realms. There, they just slow the speed of the law, rather than badly break the law; the more variability, the slower is the convergence to the truth. Black Matter Lives (BML) in the mathematical space of “in probability” or “almost surely.”

 

It is time to disclose the truth about White. We suggested White stands for “in probability,” and as such, with many repeated trials, the probability Mr. White gets caught gets closer to 1.  This is also known as the “Weak LLN.” Surely enough, then, in BB, WLLN catches on Mr. White. By being unpredictable, the chances to escape the law were momentarily increased.

 

In quantitative trading, the unescapable truth is that Black Matter Lives in a Black Box. There is commonly no telling of what happens inside those Black Boxes, except that they better do attempt to follow the Laws of Statistics. Laws of Statistics, the WLLN and SSLN, rule over the possible profitability in case you have a Long (enough) Term Capital Management horizon.

 

It is all Black and White. Kolmogorov’s “0-1 Law,” too, suggests you to establish strategies that hold water in all circumstances so that the probability of a loss approaches 0. You are not advised to devise weak strategies that do not converge to the truth with probability 1. You want the almost sure (“a.s.”) convergence enforced by Huell. Thus, you better call SLLN.

 

Disclaimer: This article is not intended to give investment advice. The author assumes no liability for mishandled investment portfolios applying the aforementioned statistical laws.

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