Related papers: Maybe there's no such thing as a random sequence

Quantum theory allows for randomness generation in a device-independent setting, where no detailed description of the experimental device is required. Here we derive a general upper bound on the amount of randomness that can be generated in…

The autocorrelation function and the run structure are two basic notions for binary sequences, and have been used as two independent postulates to test randomness of binary sequences ever since Golomb 1955. In this paper, we prove for…

Consider n unit intervals, say [1,2], [3,4], ..., [2n-1,2n]. Identify their endpoints in pairs at random, with all (2n-1)!! = (2n-1) (2n-3) ... 3 1 pairings being equally likely. The result is a collection of cycles of various lengths, and…

Outlier hypothesis testing is studied in a universal setting. Multiple sequences of observations are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are distributed…

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or…

Morphic sequences form a natural class of infinite sequences, extending the well-studied class of automatic sequences. Where automatic sequences are known to have several equivalent characterizations and the class of automatic sequences is…

We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as…

In this note we classify sequences according to whether they are morphic, pure morphic, uniform morphic, pure uniform morphic, primitive morphic, or pure primitive morphic, and for each possibility we either give an example or prove that no…

Confidence sequences are confidence intervals that can be sequentially tracked, and are valid at arbitrary data-dependent stopping times. This paper presents confidence sequences for a univariate mean of an unknown distribution with a known…

We apply a common measure of randomness, the entropy, in the context of iterated functions on a finite set with n elements. For a permutation, it turns out that this entropy is asymptotically (for a growing number of iterations) close to…

A strictly stationary sequence of random variables is constructed with the following properties: (i) the random variables take the values -1 and +1 with probability 1/2 each, (ii) every five of the random variables are independent, (iii)…

Universal outlier hypothesis testing is studied in a sequential setting. Multiple observation sequences are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are…

We present a simple randomized procedure for the prediction of a binary sequence. The algorithm uses ideas from recent developments of the theory of the prediction of individual sequences. We show that if the sequence is a realization of a…

A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of the indices. De Finetti's theorem characterizes all $\{0,1\}$-valued exchangeable sequences as a "mixture" of…

Sequential sampling occurs when the entire population is not known in advance and data are obtained one at a time or in groups of units. This manuscript proposes a new algorithm to sequentially select a balanced sample. The algorithm…

In a metric space, such as the real numbers with their standard metric, a set A is open if and only if no sequence with terms outside of A has a limit inside A. Moreover, a metric space is compact if and only if every sequence has a…

Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…

Abstract Contextuality is a property of systems of random variables. The identity of a random variable in a system is determined by its joint distribution with all other random variables in the same context. When context changes, a variable…

In the field of pattern mining, a negative sequential pattern is specified by means of a sequence consisting of events to occur and of other events, called negative events, to be absent. For instance, containment of the pattern $\langle a\…

The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a…