For a regular uncountable cardinal κ and a cardinal λ with cf(λ) < κ < λ, we investigate the consistency strength of the existence of a stationary set in ${}_{\kappa }\lambda$ which cannot be split into λ⁺ many pairwise disjoint stationary subsets. To do this, we introduce a new notion for ideals, which is a variation of normality of ideals. We also prove that there is a stationary set S in ${}_{\kappa }\lambda$ such that every stationary subset of S can be split into λ⁺ many pairwise disjoint stationary subsets. ...

We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$. The state space of $(Z,S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z,S)$ is the product of the uniform probability measure and a Gaussian distribution.

Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence $⟨S_{\alpha }:\alpha <\omega ₁⟩$ of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that $S\in L[r,⟨S{*}_{\alpha }:\alpha <\omega ₁⟩]$ whenever each $S{*}_{\alpha }$ is equal to $S_{\alpha }$ modulo nonstationary changes, or may have the weaker meaning that $S\in L[r,⟨S_{\alpha }\cap C:\alpha <\omega ₁⟩]$ for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that...

We present a general method for the extension of results about linear prediction for q-variate weakly stationary processes on a separable locally compact abelian group $G_2$ (whose dual is a Polish space) with known values of the processes on a separable subset $S_2\subseteq G_2$ to results for weakly stationary processes on $G_1\times G_2$ with observed values on $G_1\times S_2$. In particular, the method is applied to obtain new proofs of some well-known results of Ze Pei Jiang.

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C(x,y)$ defining an $\alpha$-determinantal point process (DPP). Assuming absolute integrability of the function $C_0\left(x\right)=C(o,x)$, we show that a stationary $\alpha$-DPP with kernel function $C_0\left(x\right)$ is “strongly” Brillinger-mixing, implying, among others, that its tail-$\sigma$-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch...

Contents0. Introduction............................................................................................................................................. 51. Linear operators generated by random elements.......................................................................... 62. Covariance operator of generalized random elements................................................................. 93. The space of generalized random elements of the second-order as an LVH-space.................

We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of $P_{\kappa }\left(X\right)$, where κ ⊆ X. Suppose μ < κ, ${\mu }^{<\mu }=\mu$, and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in ${\left[H\left(\lambda \right)\right]}^{\mu ⁺}$ which are internally club but not internally approachable.

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda }\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\right(\beta ):\beta <\alpha ]$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda ₀i>\lambda ₁>...>{\lambda }_{n-1}$ and ordinals $\alpha ₀,...,{\alpha }_{n-1}$ such that $\alpha =\alpha ₀+\cdots +{\alpha }_{n-1}$ and $f=f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i{\in }_{{\lambda }_i}\left({\alpha }_i\right)$. Under GCH we prove that if α < ω₂ then (i) ${}_{\omega }\left(\alpha \right)=s{\in }^{\alpha }\omega ,\omega ₁:s\left(0\right)=\omega$; (ii) if λ > cf(λ) = ω, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega ₁-closedin\alpha$; (iii) if cf(λ) = ω₁, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha$; (iv) if cf(λ) > ω₁, ${}_{\lambda }\left(\alpha \right){=}^{\alpha }\lambda$. This yields a complete characterization of the classes (α) for all...

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $𝒮$ of morphisms in a locally presentable category $𝒞$ of structures, the orthogonal class of objects...

Suppose that $\{Xn:n\ge 0\}$ is a stationary Markov chain and $V$ is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if $Y_n:=N^{-1/2}{\sum }_{n=0}^{N}V\left(X_n\right)$ converge in law to a normal random variable, as $N\to +\infty$. For a stationary Markov chain with the $L^2$ spectral gap the theorem holds for all $V$ such that $V\left(X_0\right)$ is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables $V$ for which the CLT holds...

It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint $G_{\delta }$-sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.

A notion of a wide-sense Markov process $X_t$ of order k ≥ 1, $X_t\sim WM\left(k\right)$, is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of $X_t$ is the k-dimensional process $x_t=(X_{t-k+1},...,X_t)$. The covariance structure of $X_t\sim WM\left(k\right)$ is considered in the general case and in the periodic case. In the general case it is shown that $X_t\sim WM\left(k\right)$ iff $x_t$ is a k-dimensional WM(1) process and iff the covariance function of $x_t$ has the triangular...

Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of $L_{\lambda ⁺\omega }$, with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with $⟨D^{ℳ},{\prec }^{ℳ}⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where $⟨D^{{ℳ}^{\text{'}}},{\prec }^{{ℳ}^{\text{'}}}⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{\lambda ⁺\omega }$ is exactly ${\beth }_{\delta \left(\lambda \right)}$. It was proved in [BK71] that if ℵ₀ < λ < κ$areregularcardinalnumbers,thenthereisaforcingextension,preservingcofinalities,suchthatintheextension$2λ = κ$and\delta \left(\lambda \right)<\lambda ⁺⁺.Weimprovethisresultbyprovingthefollowing:Suppose\aleph ₀<\lambda <\theta \le \kappa arecardinalnumberssuchthat$∙ ${\lambda }^{<\lambda }=\lambda$; ∙ cf(θ) ≥ λ⁺ and ${\mu }^{\lambda }<\theta$ whenever μ < θ; ∙ ${\kappa }^{\lambda }=\kappa$. Then there...

We study the flat region of stationary points of the functional ${\int }_{\Omega }F\left(\right|\nabla u\left(x\right)\left|\right)dx$ under the constraint $u\le M$, where $\Omega$ is a bounded domain in ${ℝ}^2$. Here $F\left(s\right)$ is a function which is concave for $s$ small and convex for $s$ large, and $M>0$ is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when $\Omega$ is a ball. We also analyze some other qualitative properties. Moreover,...

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space $𝙴$. For an open set $𝙶\subseteq 𝙴$, let ${\tau }_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$. Let $r:[0,\infty [\to [0,\infty [$ and $R\left(t\right)={\int }_0^{t}r\left(s\right)\mathrm{d}s$. We give necessary and sufficient conditions for ${𝔼}_{\mu }R\left({\tau }_{𝙶}\right)lt;\infty$ in terms of the behavior near the origin of the spectral measure of $-A^{𝙶}$. When $r\left(t\right)=t^l$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for ${\tau }_{𝙶}$...

Let $(X_i,i=1,2,...)$ be the normalized gaussian system such that $X_i\in N(0,1)$, i = 1,2,... and let the correlation matrix ${\rho }_{ij}=E\left(X_i{X}_j\right)$ satisfy the following hypothesis: $C=su{p}_{i\ge 1}{\sum }_{j=1}^{\infty }\left|{\rho }_{i,j}\right|<\infty$. We present Gebelein’s inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy’s norm for the gaussian sequence etc. The main result is that (f(X₁) + ⋯ + f(Xₙ))/n → 0 a.s. for f ∈ L¹(ν) with (f,1)ν = 0.

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

Let a space $X$ be Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff nonsingle point spaces $X_{\alpha }$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau$. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau }$ or ${\omega }^{\tau }$ and a cardinal $\tau$ is infinite and not countably cofinal.

Let $m\ge 3$ be an integer. The so-calledis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m\le 26$, the asymptotic behavior of the process is Gaussian, but for $m\ge 27$ it is no longer Gaussian and a limit $W^{DT}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of...

We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $(M^n,g)$ be a closed $n$-manifold, $n\ge 3$. The critical Kirchhoff systems we consider are written as $$\left(a+b\sum _{j=1}^p{\int }_M{\left|\nabla u_j\right|}^{2}d{v}_g\right){\Delta }_g{u}_i+\sum _{j=1}^p{A}_{ij}{u}_j={\left|U\right|}^{2^{☆}-2}{u}_i$$ for all $i=1,\cdots ,p$, where ${\Delta }_g$ is the Laplace-Beltrami operator, $A$ is a $C^1$-map from $M$ into the space $M_s^p\left(ℝ\right)$ of symmetric $p\times p$ matrices with real entries, the $A_{ij}$’s are the components of $A$, $U=(u_1,\cdots ,u_p)$, $\left|U\right|:M\to ℝ$ is the Euclidean norm of $U$, $2^{☆}=\frac{2n}{n-2}$ is the critical Sobolev exponent, and...

The symbol ${\left(X_I\right)}_{\kappa }$ (with κ ≥ ω) denotes the space $X_I:={\prod }_{i\in I}{X}_i$ with the κ-box topology; this has as base all sets of the form $U={\prod }_{i\in I}{U}_i$ with $U_i$ open in $X_i$ and with $|i\in I:U_i\ne X_i|<\kappa$. The symbols w, d and S denote respectively the weight, density character and Suslin number. Generalizing familiar classical results, the authors show inter alia: Theorem 3.1.10(b). If κ ≤ α⁺, |I| = α and each $X_i$ contains the discrete space 0,1 and satisfies $w\left(X_i\right)\le \alpha$, then $w\left(X_{\kappa }\right)={\alpha }^{<\kappa }$. Theorem 4.3.2. If $\omega \le \kappa \le \left|I\right|\le 2^{\alpha }$ and $X={\left(D\left(\alpha \right)\right)}^I$ with D(α) discrete, |D(α)| = α, then $d\left({\left(X_I\right)}_{\kappa }\right)={\alpha }^{<\kappa }$. Corollaries 5.2.32(a)...

Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${\left({\alpha }_j\right)}_{j=1}^{\infty }$ of scalars, there exists a subsequence ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ such that either every subsequence of ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ defines a universal series, or no subsequence of ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ defines a universal series. In particular examples we decide which of the two cases holds.

In the present work, we consider spectrally positive Lévy processes $(X_t,t\ge 0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law ${\mathbb{P}}_x^{☆}$ of this conditioned process (starting at $xgt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths,...

For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.