Related papers: The strong predictable representation property in …
We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded…
We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may…
The study of representations is of fundamental importance to any form of communication, and our ability to exploit them effectively is paramount. This article presents a novel theory -- Representational Systems Theory -- that is designed to…
A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the…
Given a sequence $(M^n)^{\infty}_{n=1}$ of nonnegative martingales starting at $M^n_0=1$, we find a sequence of convex combinations $(\widetilde{M}^n)^{\infty}_{n=1}$ and a limiting process $X$ such that…
Transformers pretrained via next token prediction learn to factor their world into parts, representing these factors in orthogonal subspaces of the residual stream. We formalize two representational hypotheses: (1) a representation in the…
When describing the effective dynamics of an observable in a many-body system, the repeated randomness assumption, which states that the system returns in a short time to a maximum entropy state, is a crucial hypothesis to guarantee that…
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…
The objective of this article is to investigate the asymptotic behavior of the persistence diagrams of a random cubical filtration as the window size tends to infinity. Here, a random cubical filtration is an increasing family of random…
In this paper we study progressive filtration expansions with cadlag processes. Using results from the weak convergence of sigma fields theory, we first establish a semimartingale convergence theorem. Then we apply it in a filtration…
A principle is modified that underlies the theory of organic fiducial inference as this theory was presented in an earlier paper. This modification, which is arguably a natural one to make, allows Bayesian inference to sometimes have a…
There is a vast literature on representation learning based on principles such as coding efficiency, statistical independence, causality, controllability, or symmetry. In this paper we propose to learn representations from sequence data by…
In the presence of monotone information, the stochastic Thiele equation describing the dynamics of state-wise prospective reserves is closely related to the classic martingale representation theorem. When the information utilized by the…
In lifelong learning, tasks (or classes) to be learned arrive sequentially over time in arbitrary order. During training, knowledge from previous tasks can be captured and transferred to subsequent ones to improve sample efficiency. We…
In Pure Inductive Logic, the principle of Strong Predicate Exchangeability is a rational principle based on symmetry that sits in between the principles of Predicate Exchangeability and Atom Exchangeability. We will show a de Finetti -…
In this paper we will give a categorical proof of the Radon-Nikodym theorem. We will do this by describing the trivial version of the result on finite probability spaces as a natural isomorphism. We then proceed to Kan extend this…
The problem of belief tracking in the presence of stochastic actions and observations is pervasive and yet computationally intractable. In this work we show however that probabilistic beliefs can be maintained in factored form exactly and…
We present a large deviation property for the pattern statistics representing the number of occurrences of a symbol in words of given length generated at random according to a rational stochastic model. The result is obtained assuming that…
We deal with various alternative decompositions of F-martingales with respect to the filtration G which represents the enlargement of a filtration F by a progressive flow of observations of a random time that either belongs to the class of…