Global randomness and local randomness are different. Most philosophical conceptions of randomness are global—because they are based on the idea that "in the long run" a sequence looks truly random, even if certain sub-sequences would not look random. In a "truly" random sequence of numbers of sufficient length, for example, it is probable there would be long sequences of nothing but zeros, though on the whole the sequence might be random. Local randomness refers to the idea that there can be minimum sequence lengths in which random distributions are approximated. Long stretches of the same digits, even those generated by "truly" random processes, would diminish the "local randomness" of a sample (it might only be locally random for sequences of 10,000 digits; taking sequences of less than 1,000 might not appear random at all, for example).
A sequence exhibiting a pattern is not thereby proved not statistically random. According to principles of Ramsey theory, sufficiently large objects must necessarily contain a given substructure ("complete disorder is impossible").
I don't quite understand the meanings of the two sentences in bold.
Does the first sentence mean that something makes a sequence local random at a longer length, and not local random at a shorter length?
How does the example inside the parenthesis work?
- Does the second sentence mean that a sequence exhibiting a pattern cannot be proved to be not statistically random? Why?