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Would a truly random function exhibit self-similarity?

By truly random I basically mean that if you have some function that produces a binary value, each new value will have exactly 50% chance of being a 0 or a 1, and will be entirely unrelated to the previous entry.

By self similarity I mean that if you made a graph over time, where the line represents the ratio of 1s to 0s at that time, it would look self similar.

And if so, it seems like it wouldn't be random at all.

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What precisely do you mean by "truly random"? And could you be explicit about the relevance of this question to statistics or machine learning? – whuber♦ Sep 14 '11 at 6:02
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...and what do you precisely mean by self-similarity? – mbq♦ Sep 14 '11 at 12:07
I edited to reflect this. – weezybizzle Sep 14 '11 at 14:24
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Your last edit reveals where you're coming from. Your question might succinctly be put, "how can random phenomena exhibit predictable properties?" You might like to read a little about laws of large numbers. – whuber♦ Sep 14 '11 at 19:07
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I think you need a more precise definition of what exactly you mean by self-similarity. Can you give a mathematical definition? – D.W. Oct 14 '11 at 20:35
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closed as not a real question by whuber♦ Mar 13 at 22:04

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.

1 Answer

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No. That's the answer. This extra bit is to get up to the required minimum number of characters!

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Because it's simple enough to create a sigma algebra over a set of self-similar functions and endow it with a uniform probability, it seems any answer has to be qualified. – whuber♦ Sep 14 '11 at 6:04
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So, does that mean that my "no" should be a qualified no? Surely self-similarity is non-randomness. There is more to randomness than uniform distribution of the digits. (And isn't an interest in randomness a sign of being a bit nerdy?) – Michael Lew Sep 15 '11 at 0:25
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Don't understand the point about nerdiness, but random digits will not create a structure that repeats itself at different scales (e.g., daily, monthly, yearly). The extensive literature on chaos theory/nonlinear dynamical systems (NDS) makes this very clear: chaotic systems can have an inherent structure even though they appear on the surface to be random. These are actually deterministic and can be definitively distinguished from random processes through various tests, including Lyapunov exponent. – rolando2 Oct 16 '11 at 17:40

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