von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

An L0（F,R）-valued Function＇s Intermediate Value Theorem and Its Applications to Random Uniform Convexity

Let (Ω , F , P ) be a probability space and L0 ( F, R ) the algebra of equivalence classes of real- valued random variables on (Ω , F , P ). When L0 ( F, R ) is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0 ( F, R ) to L0 ( F, R ). As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module ( S,|| · ||) is random uniformly convex iff Lp ( S ) is uniformly convex for each fixed positive number p such that 1 p + ∞ .     

Some results related to interpolation on hardy spaces of regular martingales

Let (Ω,F, P) be a probability space and {F _n}_n≥0 a regular increasing sequence of sub-σ-fields ofF. LetH ₁(Ω) be the usual Hardy space ofF _n-martingales. We show that the couple (H ₁(Ω),L _∞(Ω)) is a partial retract of (L ₁(Ω),L _∞(Ω)). It is also proved that (L _p(Ω),BMO(Ω)) is a partial retract of (L _p(Ω),L _∞(Ω)) for all 1<p<∞.     

Derived space of L p (G, H)

Let G denote a locally compact abelian group and H a separable Hilbert space. Let L _p (G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual L _p space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space L _p (G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then L _p ⁰ (G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let S _p (G, H) be the L ₁(G) Banach module of functions in L _p (G, H) having unconditionally convergent Fourier series in L _p -norm. We show that S _p (G, H) coincides with the derived space L _p ⁰ (G, H), as in the scalar valued case. We also show that if G is compact and abelian, then L _p ⁰ (G, H) = L ₂(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ L _p (G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in L _p -norm, then F ∈ L ₂(G, H). Let Ω be the set of all functions on Γ taking only the values 1, ?1 and Ω* be the set of all complex-valued functions on Γ having absolute value 1. As an application of the derived space L _p ⁰ (G, H), we prove the following main result of this paper. Let G be a compact abelian group and F be an H-valued function on the dual group Γ such that $$ \sum \omega (\gamma )F(\gamma )\gamma $$ is a Fourier-Stieltjes series of some measure µ ∈ M(G, H) for every scalar function ω such that |ω(γ)| = 1. Then F ∈ l ²(Γ, H).     

The relations among the three kinds of conditional risk measures

Let(Ω,E,P)be a probability space,F a sub-σ-algebra of E,Lp(E)(1 p+∞)the classical function space and Lp F(E)the L0(F)-module generated by Lp(E),which can be made into a random normed module in a natural way.Up to the present time,there are three kinds of conditional risk measures,whose model spaces are L∞(E),Lp(E)(1 p+∞)and Lp F(E)(1 p+∞)respectively,and a conditional convex dual representation theorem has been established for each kind.The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems.We first establish the relation between Lp(E)and Lp F(E),namely Lp F(E)=Hcc(Lp(E)),which shows that Lp F(E)is exactly the countable concatenation hull of Lp(E).Based on the precise relation,we then prove that every L0(F)-convex Lp(E)-conditional risk measure(1 p+∞)can be uniquely extended to an L0(F)-convex Lp F(E)-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter,which shows that the study of Lp-conditional risk measures can be incorporated into that of Lp F(E)-conditional risk measures.In particular,in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L0-convex conditional risk measures.∞     

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

On a characterization of orthogonality with respect to particular sequences of random variables in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>

This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space L ²(Ω, F, ℙ) of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of L ²(Ω, F, ℙ) to be orthogonal to some other sequence in L ²(Ω, F, ℙ). The result obtained is interesting from the point of view of the time series analysis, since it can be applied to a class of sequences random variables that exhibit a monotonically increasing variance. An application to ergodic theorem is also provided.     

Modular Lie powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L_n(V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L_n(V) as a module for the general linear group GL_r(F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L_k(V) and the indecomposable direct summands of L_n(V) which are not isomorphic to direct summands of V^⊗n. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups. Second author supported by Deutsche Forschungsgemeinschaft (DFG-Scho 799).     

GIRSANOV TRANSFORMATION AND ITS APPLICATION TO THE THEORY OF ENLARGEMENT OF FILTRATIONS

Given a complete filtered probability space (Ω, (F _t ) _{t ∈ [0,1]}, F,P). If we enlarge the filtration by random variables satisfying condition A given in [Ja] and condition B ^X which is defined in this paper (or in [FI]), then the semimartingale property also preserves. Moreover, the invariance of martingale property for Poisson martingale under a simultaneous enlargement of filtration and change of equivalent probability measure can be obtained.     

Linear groups with minimality condition for some infinite-dimensional subgroups

Let F be a field, let A be a vector space over F, and let GL(F, A) be the group of all automorphisms of the space A. If H is a subgroup of GL(F, A), then we set aug dim_F (H) = dim_F (A(ωFH)), where ωFH is the augmentation ideal of the group ring FH. The number aug dim_F (H) is called the augmentation dimension of the subgroup H. In the present paper, we study locally solvable linear groups with minimality condition for subgroups of infinite augmentation dimension. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1476–1489, November, 2005.     

Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations

In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P^′(t,ω)=A(t,ω)(1?P(t,ω))P(t,ω), t∈[t₀,T], P(t₀,ω)=P₀(ω), where ω is any outcome in the sample space Ω. In the recent contribution [Cortés, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121–138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P₀ to approximate the density function f₁(p,t) of P(t): A(t) is expressed as a Karhunen–Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P₀, and the density of P₀, , is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L²([t₀,T]×Ω), so that we include other expansions such as random power series. We only require absolute continuity for P₀, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For , only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence of density functions in terms of expectations that tends to f₁(p,t) pointwise. Numerical examples illustrate our theoretical results.     

An optimal bound on the tail distribution of the number of recurrences of an event in product spaces

 Let X ₁ ,X ₂ ,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that |X _i+1 +...+X _j |≥a for some integers 0≤i<j<∞. For each k≥2 we upper-bound the probability that A occurs k or more times, i.e. that A occurs on k or more disjoint intervals, in terms of P(A), the probability that A occurs at least once. More generally, let X=(X ₁ ,X ₂ ,...)Ω=Π _{j ≥1}Ω _j be a random element in a product probability space (Ω,ℬ,P=⊗ _{j ≥1} P _j ). We are interested in events AB that are (at most contable) unions of finite-dimensional cylinders. We term such sets sequentially searchable. Let L(A) denote the (random) number of disjoint intervals (i,j] such that the value of X _(i,j] =(X _i+1 ,...,X _j ) ensures that XA. By definition, for sequentially searchable A, P(A)≡P(L(A)≥1)=P(𝒩_{−ln (P(Ac))} ≥1), where 𝒩_γ denotes a Poisson random variable with some parameter γ>0. Without further assumptions we prove that, if 0<P(A)<1, then P(L(A)≥k)<P(𝒩_{−ln (P(Ac))} ≥k) for all integers k≥2. An application to sums of independent Banach space random elements in l ^∞ is given showing how to extend our theorem to situations having dependent components. Received: 8 June 2001 / Revised version: 30 October 2002 Published online: 15 April 2003 RID="*" ID="*" Supported by NSF Grant DMS-99-72417. RID="†" ID="†" Supported by the Swedish Research Council. Mathematics Subject Classification (2000): Primary 60E15, 60G50 Key words or phrases: Tail probability inequalities – Hoffmann-Jo rgensen inequality – Poisson bounds – Number of event recurrences – Number of entrance times – Product spaces     

Rates of convergence for classes of functions: The non-i.i.d. case

Let X_i, i ≥ 1, be a sequence of φ-mixing random variables with values in a sample space (X, A). Let L(X_i) = P_(i) for all i ≥ 1 and let _n, n ≥ 1, be classes of real-valued measurable functions on (X, A). Given any function g on (X, A), let S_n(g) = Σ_{i = 1}ⁿ {g(X_i) − Eg(X_i)}. Under weak metric entropy conditions on _n and under growth conditions on both the mixing coefficients and the maximal variance V V(n) max_{i ≤ n} sup_{g n} ∫ g² dP_(i), we show that there is a numerical constant U < ∞ such that

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a.s. ^*, where _{i = 1}^xP_(i) and H H(n) is the square root of the entropy of the class _n. Additionally, the rate of convergence H⁻¹(n/V)^1/2 cannot, in general, be improved upon. Applications of this result are considered.     

Uniform bound in the central limit theorem for Banach space valued dependent random variables

Let F be a Banach space with a sufficiently smooth norm. Let (X_i)_i≤n be a sequence in L_F², and T be a Gaussian random variable T which has the same covariance as X = Σ_i≤nX_i. Assume that there exists a constant G such that for s, δ≥0, we have P(sTs+δ)Gδ. (*) We then give explicit bounds of Δ(X) = sup_i|P(|X|≤t)−P(|T|≤t)| in terms of truncated moments of the variables X_i. These bounds hold under rather mild weak dependence conditions of the variables. We also construct a Gaussian random variable that violates (^*).     

Nonlinear random ergodic theorems for affine operators

Let (Ω,ß,μ) be a finite measure space and let (S,F,ν) be another probability measure space on which a measure preserving transformation φ is given. We introduce the so-called affine systems and prove a vector-valued nonlinear random ergodic theorem for the random affine system determined by a strongly F-measurable family of affine operators, where B is a reflexive Banach space, is a strongly F-measurable family of linear contractions on L₁(Ω,B) as well as on L_∞(Ω,B) and ξ is a function in (I−T)Lp(S×Ω,B) (1?p<∞) with the operator T defined by Tf(s,ω)=[Tsfφs](ω) which denotes the F⊗ß-measurable version of Tsfφs(ω). Moreover, some variant forms of the nonlinear random ergodic theorem are also obtained with some examples of affine systems for which the nonlinear ergodic theorems fail to hold.     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Orlicz-Hardy spaces associated with operators

Let L be a linear operator in L2(Rn) and generate an analytic semigroup {e-tL}t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t-1/ω-1(t-1) for t ∈ (0, ∞). We introduce the Orlicz-Hardy space Hω, L(Rn) and the BMO-type space BMOρ, L(Rn) and establish the John-Nirenberg inequality for BMOρ, L(Rn) functions and the duality relation between Hω, L(Rn) and BMOρ, L...     

SOME MULTIPLIER THEOREMS FOR ANISOTROPIC HARDY SPACES ——In Memory of Professor Yongsheng Sun

Let A be a symmetric expansive matrix and Hp(Rn) be the anisotropic Hardy space associated with A. For a function m in L∞(Rn), an appropriately chosen function η in Cc∞(Rn) and j ∈ Z define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 ＜ p ＜ 1 and (m)j belongs to the anisotropic nonhomogeneous Herz space K11/p-1,p(Rn), then m is a Fourier multiplier from Hp(Rn) to Lp(Rn). For p = 1, a similar result is obtained if the space K10,1(Rn) is replaced by a slightly smaller space K(w).Moreover, the authors show that if 0 ＜ p ≤ 1 and if the sequence {(mj)V} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp(Rn) to Lp(Rn).