Randomness Formal definition

Question

What is the most accurate definition of randomness? I saw some posts concerning random variables, but I would say I don't intuitively think they present some kind of randomness, given that they have certain distributions and density functions which for many realizations result in certain patterns.

Answer 1

"Random" is used in different senses in various mathematical and scientific contexts. I don't believe that anyone can claim that there is one most accurate definition that is appropriate for all of them.

You're of course right about "random variable". It is a technical term, and the outcomes of a random variable don't have to exhibit "randomness" in the sense that you probably have in mind. One way to define "random variable" is: a function that maps outcomes in a probability space to real numbers. (There are other definitions; some are more general.) Another way to think of it is that a random variable is an abstraction that assigns probabilities to sets of numbers.

Consider the random variable $X$ that always returns the integer 0. This is a random variable that assigns the probability 1 to 0, and probability zero to all other numbers. If you repeatedly look at $X$ (with independent trials), you will get a sequence of numbers $0, 0, 0, 0, 0, \ldots$ . That does not seem very "random". So to specify what it means for a sequence to be random, we need other ideas.

For a sense of "randomness" of sequences of numbers, the link in J. W. Tanner's comment does look good. It's primarily about what's often called "algorithmic randomness".` Li and Vitanyi's An Introduction to Kolmogorov Complexity and Its Applications is a standard, comprehensive introduction to this subject. A new edition just came out. Calude's Information and Randomness: An Algorithmic Perspective is written at a similar level, I feel. It doesn't try to do as much, I think, which might be an advantage. Chaitin's Algorithmic Information Theory might be a little easier, even though the choice of topics and the presentation are somewhat idiosyncratic.

Your question, "What is the most accurate definition", might be about algorithmic randomness. The problem with answering is that there are several answers that are equivalent. However, there are different, closely-related, non-equivalent definitions as well. For finite strings of numbers, for example, there are definitions in terms of constraints on Turing machines. (The general idea here is that "more random" means "requires a longer program to generate it", but this can also be specified in other ways, so that "random" means something like "passes all possible statistical tests for patternlessness".) These ideas can be extended for infinite sequences. (EDIT: For infinite sequences, there are in fact stronger and weaker variations on this general kind of idea. See the textbooks by Nies or by Downey and Hirschfeldt.)

The algorithmic randomness literature is about characterizing a sequence as random or not, or as more or less random. But you might also think of some physical events or processes as random in a different sense, whether or not what's produced is a sequence of outcomes that is random in algorithmic sense. One might call coin tossing random, for example, even though some of the outcomes that are possible in a sequence of coin tosses are not algorithmically random. If you toss a fair coin 10,000 times, one possibility is that, just by chance, you will get heads, tails, heads, tails, heads, tails, ..., alternating 5000 heads with 5000 tails. (Here I really mean flipping a physical coin in the air. "Coin tossing" is not just a concrete metaphor here for something purely mathematical.) If it was possible to toss a fair coin an infinite number of times, it would be possible to get all heads, or all tails, or alternating heads and tails ad infinitum. (To deny that is to deny a fundamental assumption about coin tossing.)

You can represent and reason about such physical, chancy processes like coin tossing using mathematical random variables, but as before, being representable by a random variable doesn't imply that the underlying process is random in an intuitive sense. (The random variable $X$ mentioned above could be used to represent the following physical procedure: Place a coin on a table, with tails up. After that, don't move the coin or the table. Once per minute, look at the coin to see whether it shows heads or tails. If it shows tails, write down $0$. Otherwise write down the value of $\pi$.)

For a philosophical discussion of different senses of randomness, Eagle's "Chance verus Randomness" article in the Stanford Encyclopedia of Philosophy could be worth reading. Hájek's discussion of frequency interpretations in the article on "Interpretations of Probability" could also be illuminating.