Related papers: Conditional normality and finite-state dimensions …
One can consider $\mu$-Martin-L\"of randomness for a probability measure $\mu$ on $2^{\omega}$, such as the Bernoulli measure $\mu_p$ given $p \in (0, 1)$. We study Bernoulli randomness of sequences in $n^{\omega}$ with parameters $p_0,…
We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense…
In the theory of algorithmic randomness, several notions of random sequence are defined via a game-theoretic approach, and the notions that received most attention are perhaps Martin-Loef randomness and computable randomness. The latter…
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure…
The conventional definition of extremality of a finite collection of sets is extended by replacing a fixed point (extremal point) in the intersection of the sets by a collection of sequences of points in the individual sets with the…
It is well known that normality can be described as incompressibility via finite automata. Still the statement and the proof of this result as given by Becher and Heiber (2013) in terms of "lossless finite-state compressors" do not follow…
Randomness is fundamental in quantum theory, with many philosophical and practical implications. In this paper we discuss the concept of algorithmic randomness, which provides a quantitative method to assess the Borel normality of a given…
We define a notion of randomness for individual and collections of formal languages based on automatic martingales acting on sequences of words from some underlying domain. An automatic martingale bets if the incoming word belongs to the…
A real \alpha is called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to \alpha. It is known that the randomness of an r.e. real \alpha can be characterized in…
In the first part of this paper the notion of natural metric on the set of natural numbers is defined. It is such metric that the completion of N is a compact metric space that a probability borel measure exists in order that the sequence…
In this paper, we introduce and develop the concept of conditional quantization for Borel probability measures on $\mathbb{R}^k,$ considering both constrained and unconstrained frameworks. For each setting, we define the associated…
Since human randomness production has been studied and widely used to assess executive functions (especially inhibition), many measures have been suggested to assess the degree to which a sequence is random-like. However, each of them…
An infinite binary sequence is deemed to be random if it has all definable properties that hold almost surely for the usual probability measure on the set of infinite binary sequences. There are only countably many such properties, so it…
The definition of the conditional probability is very important in the theory of the probability. This definition is based on the fact, that random events can be simultaneously measurable. This paper deal with the problem of conditioning…
We consider a conception of reality that is the following: An object is 'real' if we know that if we would try to test whether this object is present, this test would give us the answer 'yes' with certainty. If we consider a conception of…
Algorithmic theories of randomness can be related to theories of probabilistic sequence prediction through the notion of a predictor, defined as a function which supplies lower bounds on initial-segment probabilities of infinite sequences.…
A prefix normal word is a binary word whose prefixes contain at least as many 1s as any of its factors of the same length. Introduced by Fici and Lipt\'ak in 2011 the notion of prefix normality is so far only defined for words over the…
Characterizing inhomogeneous temporal patterns in natural and social phenomena is important to understand underlying mechanisms behind such complex systems, hence even to predict and control them. Temporal inhomogeneities in event sequences…
The subject of this paper is regularity-preserving aggregation of regular norms on finite-dimensional linear spaces. Regular norms were introduced in [5] and are closely related to ``type 2'' spaces [9, Chapter 9] playing important role in…
For a fixed alphabet A, an infinite sequence X is said to be normal if every word w over A appears in X with the same frequency as any other word of the same length. A classical result relates normality to finite automata as follows: a…