Related papers: Conditional normality and finite-state dimensions …
Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters…
We investigate the behavior of the periods and border lengths of random words over a fixed alphabet. We show that the asymptotic probability that a random word has a given maximal border length $k$ is a constant, depending only on $k$ and…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define several notions of randomness associated with interval, rather than precise,…
In 2004, Dai, Lathrop, Lutz, and Mayordomo defined and investigated the finite-state dimension (a finite-state version of algorithmic dimension) of a sequence $S \in \Sigma^\infty$ and, in 2018, Case and Lutz defined and investigated the…
There are several ways to measure the compressibility of a random measure; they include general approaches such as using the rate-distortion curve, as well as more specific notions, such as the Renyi information dimension (RID). The RID…
We show how to calculate the finite-state dimension (equivalently, the finite-state compressibility) of a saturated sets $X$ consisting of {\em all} infinite sequences $S$ over a finite alphabet $\Sigma_m$ satisfying some given condition…
A fundamental problem with the Nash equilibrium concept is the existence of certain "structurally deficient" equilibria that (i) lack fundamental robustness properties, and (ii) are difficult to analyze. The notion of a "regular" Nash…
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 concept of boolean autonomous deterministic regular asynchronous system has its origin in switching theory, the theory of modeling the switching circuits from the digital electrical engineering. The attribute boolean vaguely refers to…
Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program…
A new definition of random sets is proposed. It is based on the distance in measurable space and uses negative definite kernels for continuation from initial space to that of random sets. This approach has no connection to Hausdorff…
We introduce multi-head finite-state dimension, a generalization of finite-state dimension in which a group of finite-state agents (the heads) with oblivious, one-way movement rules, each reporting only one symbol at a time, enable their…
The concept of a variance on a category is introduced as a two-sided strict factorization system. By employing variances, we define functors of variance in a more general setting than is usually considered, thereby eliminating the need for…
Suppose you look at today's stock prices and bet on the value of the first digit. One could guess that a fair bet should correspond to the frequency of $1/9 = 11.11%$ for each digit from 1 to 9. This is by no means the case, and one can…
QBism regards quantum mechanics as an addition to probability theory. The addition provides an extra normative rule for decision-making agents concerned with gambling across experimental contexts, somewhat in analogy to the double-slit…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
We extend a general result showing that the asymptotic behavior of high moments, factorial or standard, of random variables, determines the asymptotically normality, from the one dimensional to the multidimensional setting. This approach…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
De Finetti theorems tell us that if we expect the likelihood of outcomes to be independent of their order, then these sequences of outcomes could be equivalently generated by drawing an experiment at random from a distribution, and…
This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a…