Sigmasidosten

Sigmasidosten is a theoretical concept in probability theory used to describe the aggregated influence of a sequence of sigma-algebras on a random variable. In a probability space (Ω, F, P) with a filtration {F_t}t≥0 and an integrable X, one can form conditional increments Δ_t = E[X | F_t] − E[X | F_{t−1}] and attach a weight sequence α_t determined by a function σ. The sigmasidosten of X with respect to the filtration is the limit, when it exists, of the weighted partial sums S_T = ∑_{t=1}^T α_t Δ_t. When the limit exists in L^1 or in probability, Sig(X; {F_t}) is a random variable that encodes the cumulative effect of new information arriving through the filtration. The construction can be viewed as a generalization of how a martingale accumulates conditional information, extending ideas found in decompositions by separating the influence of information flow from other components.

In practice, sigmasidosten serves as a theoretical tool in discussions of information flow, convergence of conditional expectations, and decompositions of stochastic processes. It is related to, but distinct from, standard martingale decompositions, predictable compensators, and information measures that arise from conditional expectations.

Origin and usage notes: the term is not part of standard texts and appears primarily in specialized theoretical discussions where researchers explore generalized information accumulation across filtrations. Its precise formulation and applicability can vary among authors, and it is often understood as a heuristic or exploratory construct rather than a universally fixed object. See also conditional expectation, filtration, martingale, Doob decomposition.