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The term Bernoulli sequence is often used informally to refer to a realization of a Bernoulli process. However, the term has an entirely different formal definition as given below. Suppose a Bernoulli process formally defined as a single random variable (see preceding section). For every infinite sequence x of coin flips, there is a sequence of ...
The infinite words, or ω-words, can likewise be viewed as functions from to Σ. The set of all infinite words over Σ is denoted Σ ω. The set of all finite and infinite words over Σ is sometimes written Σ ∞ or Σ ≤ω. Thus an ω-language L over Σ is a subset of Σ ω.
In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). [1] If such a limit exists and is finite, the sequence is called convergent . [ 2 ]
Then, in the same way, it picks a triplet that starts with the second and third words in the generated text, and that gives a fourth word. It adds the fourth word, then repeats with the third and fourth words, and so on. [32] Random walks based on integers and the gambler's ruin problem are examples of Markov processes.
Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value (called the limit of the sequence), and they become and remain arbitrarily close to , meaning that given a real number greater than zero, all but a finite number of the elements of the sequence have a distance from less than .
A sequence can also have an infinite limit: as , the sequence (). This direct definition is easier to extend to one-sided infinite limits. While mathematicians do talk about functions approaching limits "from above" or "from below", there is not a standard mathematical notation for this as there is for one-sided limits.
Let be "0" and be "01". Now = (the concatenation of the previous sequence and the one before that).. The infinite Fibonacci word is the limit , that is, the (unique) infinite sequence that contains each , for finite , as a prefix.
Suppose (X n) is a sequence of random variables with Pr(X n = 0) = 1/n 2 for each n. The probability that X n = 0 occurs for infinitely many n is equivalent to the probability of the intersection of infinitely many [X n = 0] events. The intersection of infinitely many such events is a set of outcomes common to all of them.