Related papers: A Nonmeasurable Set from Coin Flips

Suppose we observe an infinite series of coin flips $X_1,X_2,\ldots$, and wish to sequentially test the null that these binary random variables are exchangeable. Nonnegative supermartingales (NSMs) are a workhorse of sequential inference,…

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…

Outer measures can be used for statistical inference in place of probability measures to bring flexibility in terms of model specification. The corresponding statistical procedures such as Bayesian inference, estimators or hypothesis…

It is shown that some convolution semigroups of infinitely divisible measures are invariant under the random integral mappings $I^{h,r}_{(a,b]}$ defined in $(\star)$ below. The converse implication is specified for the semigroups of…

Spaces of quasi-invariant measures supplied with different topologies are studied. Their embeddings, projective decompositions, conditions for their metrizability are investigated. Theorems about convergence of nets of quasi-invariant…

Some nondifferentiable quantities (for example, the metric signature) can be the independent physical degrees of freedom. It is supposed that in quantum gravity these degrees of freedom can fluctuate. Two examples of such quantum…

We introduce a contextual quantum system comprising mutually complementary observables organized into two or more collections of pseudocontexts with the same probability sums of outcomes. These pseudocontexts constitute non-orthogonal bases…

We propose partial measurements as a conceptual tool to understand how to operate with counterfactual claims in quantum physics. Indeed, unlike standard von Neumann measurements, partial measurements can be reversed probabilistically. We…

In this work, we establish a nontrivial level of distribution for densities on $\{1,\ldots, N\}$ obtained by a biased coin convolution. As a consequence of sieving theory, one then derives the expected lower bound for the weight of such…

We study the existence of non-separable compact spaces that support a measure and are small from the topological point of view. In particular, we show that under Martin's axiom there is a non-separable compact space supporting a measure…

It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio…

We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.

Two observables are called complementary if preparing a physical object in an eigenstate of one of them yields a completely random result in a measurement of the other. We investigate small sets of complementary observables that cannot be…

This brief paper develops a probability density that models processes for which the physical mechanism is unknown. It has desirable properties which are not realized by densities derived from Gaussian process or other classic methods. In…

Measurable sets are defined as those locally approximable, in a certain sense, by sets in the given algebra (or ring). A corresponding measure extension theorem is proved. It is also shown that a set is locally approximable in the mentioned…

Little effort has been devoted to studying generalised notions or models of (un)predictability, yet is an important concept throughout physics and plays a central role in quantum information theory, where key results rely on the supposed…

This work proposes a view of probability as a relative measure rather than an absolute one. To demonstrate this concept, we focus on finite outcome spaces and develop three fundamental axioms that establish requirements for relative…

An urn scheme is a probabilistic model in which balls are placed into urns sequentially and independently of each other. All balls share the same probability distribution for hitting the urns. In the simplest case, there is a finite number…

The existence of incompatibility is one of the most fundamental features of quantum theory, and can be found at the core of many of the theory's distinguishing features, such as Bell inequality violations and the no-broadcasting theorem. A…

We study a positive-definite function associated to a measure-preserving equivalence relation on a standard probability space and use it to measure quantitatively the proximity of subequivalence relations. This is combined with a recent…