Let C k n n nt denote the set of all k element combinations of the multiset consisting of ni occurrences of i for i t Each combination is itself a multiset For example C ff g f g f g f gg We show that multiset combinations can be listed so that successive combina tions di er by one element Multiset combinations simultaneously generalize for which minimal change listings called Gray codes are kn...

Let H be the closed real linear hull of {Xs : s ∈ R} in L2(Ω,F , P ). Then H is a real Hilbert space with inner product (Y1, Y2) := E[Y1Y2] and norm ‖Y ‖ := (Y, Y ). For I ⊂ R, put HI for the closed subspace of H spanned by {Xs : s ∈ I}, and PI for the orthogonal projection operator of H onto HI . We write P⊥ I for the projection operator onto the orthogonal complement H ⊥ I of HI , i.e., P⊥ I ...

Consider a sequence of examples (xt, yt) for t = 1, . . . , T where xt ∈ R and yt ∈ [K], where the goal of a Learner is to predict the class yt from the input xt. In the more common full-information setting, the Learner observes the true class yt after making her prediction ŷt. In the present open problem, however, we will consider the so-called bandit setting: after predicting ŷt, the Learner ...

Definition 1.1. A Gaussian process {Xt }t∈T indexed by a set T is a family of (real-valued) random variables Xt , all defined on the same probability space, such that for any finite subset F ⊂ T the random vector XF := {Xt }t∈F has a (possibly degenerate) Gaussian distribution; if these finitedimensional distributions are all non-degenerate then the Gaussian process is said to be nondegenerate....

Here we show that, in the model considered in Section 5.1 in the text, Lemmas 1-2 and Proposition 1 remain valid.1 Recall that period t output is now f(Xt + xt) = f(Xt+1), while all other aspects of the model are unchanged. The novice’s outside option remains equal to 1 1−δf (Xt) . 2 Other than the modifications listed below, the proofs of Lemmas 1-2 and Proposition 1 are identical to before. (...

The MG-SS for 0 < a < 1, a = 1 and a > 1 are illustrated in Figure 4. When 0 < a < 1, the conditional distribution p(yt|xt) is skewed towards the current unnormalized density value f(xt). The conditional draw of p(xt+1|yt) encourages taking samples with smaller density value (small moves), within the domain of the slice interval X. On the other hand, when a > 1, draws of yt tend to take smaller...

A continuous-state population-size-dependent branching process {Xt } is a modification of the Jiřina process. We prove that such a process arises as the limit of a sequence of suitably scaled population-size-dependent branching processes with discrete states. The extinction problem for the population Xt is discussed, and the limit distribution of Xt/t obtained when Xt tends to infinity.

We consider a class of subcritical superprocesses (Xt)t≥0 with general spatial motions and branching mechanisms. study the asymptotic behaviors Qt,r, distribution Xt conditioned on Xt+r not being null measure. first give existence limt→∞Qt,r limr→∞Qt,r, then show that an LlogL-type condition is equivalent to iterated limits: limr→∞limt→∞Qt,r limt→∞limr→∞Qt,r. Finally, when holds, we those limit...