Let S be a Borel subset of Polish space and F the set bounded functions f:S→R. an(·)=P(Xn+1∈·∣X1,…,Xn) n-th predictive distribution corresponding to sequence (Xn) S-valued random variables. If is conditionally identically distributed, there probability measure μ on such that ∫fdan⟶a.s.∫fdμ for all f∈F. Define Dn(f)=dn∫fdan−∫fdμ f∈F, where dn>0 constant. In this note, it shown that, under som...

There is a finite time horizon [0, T ], a probability space (Ω,F ,P), and a filtration F = {Ft}t=0 satisfying the usual conditions, with F0 = {∅,Ω} and FT = F . Each ω ∈ Ω represents a complete description of what happens from time 0 to T . Each A ∈ Ft represents an event distinguishable at time t, i.e., you know whether or not ω ∈ A at time t. The measure P is the subjective probability measur...

Evaluation of MT evaluation measures is limited by inconsistent human judgment data. Nonetheless, machine translation can be evaluated using the well-known measures precision, recall, and their average, the F-measure. The unigrambased F-measure has significantly higher correlation with human judgments than recently proposed alternatives. More importantly, this standard measure has an intuitive ...

Given a rotation invariant measure in Rn, we define the maximal operator over circular sectors. We prove that it is of strong type (p, p) for p > 1 and we give necessary and sufficient conditions on the measure for the weak type (1, 1) inequality. Actually we work in a more general setting containing the above and other situations. Let X be a topological space and μ be a Borel measure on X. By ...

We prove new separability results about free groups. Namely, if H1,…,Hk are infinite index, finitely generated subgroups of a non-abelian group F, then there exists homomorphism onto some alternating f:F↠Am such that whenever Hi is not conjugate into Hj, f(Hi) f(Hj). The proof probabilistic. count the expected number fixed points f(Hi)'s and their under carefully constructed measure.

It is proved that there exists a Sierpiński-Zygmund function, which is measurable with respect to a certain invariant extension of the Lebesgue measure on the real line R. Let E be a nonempty set and let f : E → R be a function. We say that f is absolutely nonmeasurable if f is nonmeasurable with respect to any nonzero σ-finite diffused (i.e., continuous) measure μ defined on a σ-algebra of sub...

Extending the concept of level measure μ({f⩾a}) we introduce a generalized based on family conditional aggregations operators. As recent research direction, aggregation operators extend in order to model dependence function set. Generalized measures then provide new insight into behaviour level-dependent (monotone) and related processes. We investigate detail several basic properties measure, i...

Let A be a bounded subset of R. We give an upper bound on the volume of the symmetric difference of A and f(A) where f is a translation, a rotation, or the composition of both, a rigid motion. The volume is measured by the d-dimensional Hausdorff measure, which coincides with the Lebesgue measure for Lebesgue measurable sets. We bound the volume of the symmetric difference of A and f(A) in term...

A. Probability spaces, random variables. (Shreve, Chapter 1) In this course, risk-neutral pricing theory is formulated in the language ofmeasuretheoretic probability. To the aspiring quant, measure-theoretic probability might at first appear mysterious, overly abstract, and irrelevant to quantitative analysis. However, it provides a general and powerful way to express both the conceptual ideas ...

A finite function f is a mapping of {1 , 2 ,. .. , m } into {1 , 2 ,. .. , m } ∪ { # } where # is a symbol to be thought of as ''undefined.'' This paper defines a measure M(f) of the difficulty of inverting a finite function f, which is given by M(f) = MIN    log 2 C(f) log 2 C(g) _ _________ : g an inverse of f    where C(f) is a circuit complexity measure of the difficulty of computing ...