The market’s movements are therefore beyond the “complexity horizon” of human forecasters

From A Mathematician Plays the Stock Market by John Allen Paulos.  

Returning to stocks, different market theorists will have different ideas about the likely pattern of 0s and 1s (downs and upticks) that can be expected. Strict random walk theorists are likely to believe that sequences like the third characterize price movements and that the market’s movements are therefore beyond the “complexity horizon” of human forecasters (more complex than we, or our brains, are, were we  expressed as sequences of 0s and 1s). Technical and fundamental analysts might be more inclined to believe that sequences like the second characterize the market and that there are pockets of order amidst the noise. It’s hard to imagine anyone believing that price movements follow sequences as regular as the first except, possibly, those who send away “only $99.95 for a complete set of tapes that explain this revolutionary system.

I reiterate that this approach to stock price movements is rather stark, but it does nevertheless “locate” the debate. People who believe there is some pattern to the market, whether exploitable or not, will believe that its movements are characterized by sequences of complexity somewhere between those of type two and type three above.

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In any case, there is no reason why the complexity of price movements as well as the complexity of investor/computer blends cannot change over time. The more inefficient the market is, the smaller the complexity of its price movements, and the more likely it is that tools from technical and fundamental  analysis will prove useful. Conversely, the more efficient the market is, the greater the complexity of price movements, and the closer the approach to a completely random sequence of price changes.

Outperforming the market requires that one remain on the cusp of our collective complexity horizon. It requires faster machines, better data, improved models, and the smarter use of mathematical tools, from conventional statistics to neural nets (computerized learning networks, the connections between the various nodes of which are strengthened or weakened over a period of training). If this is possible for anyone or any group to achieve, it’s not likely to remain so for long.