The distribution of the likelihood ratio for additive processes

Let {X(t), 0 ≤ t ≤ T} and {Y(t), 0 ≤ t ≤ T} be two additive processes over the interval [0, T] which, as measures over D[0, T], are absolutely continuous with respect to each other. Let μ_X and μ_Y be the measures over D[0, T] determined by the two processes. The characteristic function of $\text{ln}\text{(}\text{dμ}{}_{\mathrm{X}}\text{dμ}{}_{\mathrm{Y}}\text{)}$ with respect to μ_Y is obtained in terms of the determining parameters of the two processes.     

Mutual information in Gaussian channels

In the Gaussian channel Y(t) = Φ(t) + X(t) = message + noise, where Φ(t) and X(t) are mutually independent, the information I(Y, Φ) is evaluated. One of the results is that I(Y, Φ) < ∞ if and only if Φ ? $\text{H}$(X) = the reproducing kernel Hilbert space for X(·). And the causal formula of I(Y, Φ) is given.     

Canonical forms of Borel-measurable mappings Δ: [ω]ω → R

We show that for every Borel-measurable mapping Δ: [ω]^ω → $\text{R}$ there exists A ∈ [ω]^ω and there exists a continuous mapping Γ: [A]^ω → [A]^?ω with Γ(X) ? X such that for all X, Y ∈ [A]^ω it follows that Δ(X) = Δ(Y) if Γ(X) = Γ(Y). In a sense, this is generalization of the Erdös-Rado canonization theorem     

Some generalizations of the first Fredholm theorem to multivalued A-proper mappings with applications to nonlinear elliptic equations

Let X and Y be real normed spaces with an admissible scheme Γ = {E_n, V_n; F_n, W_n} and T: X → 2^YA-proper with respect to Γ such that dist(y, A(x)) < kc(∥ x ∥) for all y in T(x) with ∥ x ∥ ? R for some R > 0 and k > 0, where c: R⁺ → R⁺ is a given function and A: X → 2^Y a suitable possibly not A-proper mapping. Under the assumption that either T or A is odd or that (u, Kx) ? 0 for all u in T(x) with $\text{∥ x ∥ ? r > 0}\text{and some}\text{K: X → Y}{}^1$, we obtain (in a constructive way) various generalizations of the first Fredholm theorem. The unique approximation-solvability results for the equation T(x) = f with T such that T(x) ? T(y) ?A(x ? y) for x, y in X or T is Fréchet differentiable are also established. The abstract results for A-proper mappings are then applied to the (constructive) solvability of some boundary value problems for quasilinear elliptic equations. Some of our results include the results of Lasota, Lasota-Opial, Hess, Ne?as, Petryshyn, and Babu?ka.     

Sums of commutators in non-commutative Banach function spaces

Let M be a II_∞-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X(M,τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form I_X(M,τ)=M∩X(M,τ) that are contained in L^p(M,τ) for some p>0. It is proved that an element T in I_X(M,τ) is a finite sum of commutators of the form [A,B] with A∈I_X(M,τ) and B∈M if and only if the function belongs to X, where ν_T is the Brown spectral measure of T and t→λ_t(T) is the non-increasing rearrangement of the function λ→|λ| with respect to ν_T. This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L^1,∞(M,τ) is always zero.     

Equivalence of measures for some class of Gaussian random fields

We consider two Gaussian measures P₁ and P₂ on (C(G), $\text{B}$) with zero expectations and covariance functions R₁(x, y) and R₂(x, y) respectively, where R_ν(x, y) is the Green's function of the Dirichlet problem for some uniformly strongly elliptic differential operator A^(ν) of order $\text{2m, m ≥ [}\text{d}\text{2}\text{] + 1}$, on a bounded domain G in $\text{R}$^d (ν = 1, 2). It is shown that if the order of A⁽²⁾ ? A⁽¹⁾ is at most $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are equivalent, while if the order is greater than $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are not always equivalent.     

Construction of translation planes from t-spread sets

A t-spread set [1] is a set $\text{C}$ of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = q^t+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of $\text{C}$. The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let $\text{G}$ be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|a^t+1, and det (G ? H) ≠ 0 if G and H are distinct elements of $\text{G}$. Let A₁, A₂, …, A_n?GL(t + 1, q) such that det(A_i ? G) ≠ 0 for i = 1, …, n and all G?G, and det(A_i ? A_jG) ≠ 0 for i > j and all G?G. Let $\text{C = \&{0\&} ∪ G ∪ A}{}_1\text{G ∪ … ∪ A}{}_{\mathrm{n}}\text{G}$, and ∥C∥ = q^t+1. Then $\text{C}$ is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of $\text{C}$ with x = eM(x). Theorem: Let$\text{C}$be a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: For$\text{C}$constructed as aboveG ? {M(x) | x?N_λ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥N_λ∥ = 6, and ∥N_μ∥ = 4. This system coordinatizes a new projective plane.     

Estimation for the change point of volatility in a stochastic differential equation

We consider a multidimensional Itô process Y=(Y_t)_t∈[0,T] with some unknown drift coefficient process b_t and volatility coefficient σ(X_t,θ) with covariate process X=(X_t)_t∈[0,T], the function σ(x,θ) being known up to θ∈Θ. For this model, we consider a change point problem for the parameter θ in the volatility component. The change is supposed to occur at some point t^∗∈(0,T). Given discrete time observations from the process (X,Y), we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit theorems of the asymptotically mixed type.     

Calculation of the Shannon information

Let ($\text{Ω, β, μ}{}_{\mathrm{X}}$) and $\text{(?, F, μ}{}_{\mathrm{N}}\text{)}$ be probability spaces, with $\text{f: Ω × ? ? ?}\text{a}\text{β × F|F}$ measurable map. Define μ_XY on β × $\text{F}$ by μ_XY(A) = μ_X ? μ_N{(x, y): (x, f(x, y)) ?A}, and let μ_Y = (μ_X ? μ_N)^of^?1. An expression is determined for computing the Shannon information in the measure μ_XY. This expression is used to compute the information for the non-linear additive Gaussian channel, and can be used to solve the channel capacity problem.     

A general approach to optimal control of a regression experiment

Let $\text{θ}\text{θ}\text{?}$ = ($\text{θ}\text{θ}\text{?}$₁, $\text{θ}\text{θ}\text{?}$₂, …, $\text{θ}\text{θ}\text{?}$_n)′ be the least-squares estimator of θ = (θ₁, θ₂, …, θ_n)′ by the realization of the process y(t) = Σ_{k = 1}ⁿθ_kf_k(t) + ξ(t) on the interval T = [a, b] with f = (f₁, f₂, …, f_n)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and D_s-optimality are used as criteria for choosing f in X. A-, D-, and D_s-optimal models can be constructed explicitly by means of r.     

Existence and asymptotic behavior of a functional differential equation in Banach space

Existence and asymptotic behavior of solutions are given for the equation u′(t) = ?A(t)u(t) + F(t,u_t) (t ? 0) and u₀ = ? ? C([?r,0]; X)  C. The space X is a Banach space; the family $\text{{A(t) ¦ 0 ? t ? T}}$ of unbounded linear operators defined on D(A) ? X → X generates a linear evolution system and F: C → X is continuous with respect to a fractional power of A(t₀) for some t₀ ? [0, T].     

On Ulyanov inequalities in Banach spaces and semigroups of linear operators

Let X,Y be Banach spaces and {T(t):t≥0} be a consistent, equibounded semigroup of linear operators on X as well as on Y; it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y:T(2t)f_Y(t)T(t)f_X. Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X,D_X((-A)^α)) and (Y,D_Y((-A)^α)),α>0, where A is the infinitesimal generator of {T(t)}. Particular choices of X,Y are Lorentz–Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case , are partly covered.     

Distribution and characteristic functions for correlated complex Wishart matrices

Let A(t) be a complex Wishart process defined in terms of the M×N complex Gaussian matrix X(t) by A(t)=X(t)X(t)^H. The covariance matrix of the columns of X(t) is Σ. If X(t), the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t₁, t₂, where t₁<t₂. In particular, we derive the following results: the joint density of the elements of A(t₁), A(t₂), the joint density of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂), the characteristic function of the elements of A(t₁), A(t₂), the characteristic function of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂). In addition, we give the characteristic functions of the eigenvalues of a central and non-central complex Wishart, and some applications of the results in statistics, engineering and information theory are outlined.     

Unbounded operators: Perturbations and commutativity problems

Let H be a complex Hilbert space, and let G_i, i = 1, 2, be closed and orthogonal subspaces of the product space H × H. The subspace G = G₁ ⊕ G₂ is called a (graph) perturbation. We give conditions for invariance of regular operators (R.O.) under graph perturbations: When is the perturbation of a R.O. again a R.O.? If N is a Hilbert space we consider R.O. (i.e., densely defined and closed operators T) in H=$\text{L}$(N) such that G(T)=G(S)⊕V$\text{H}$(M, where G denotes the graph, S is a decomposable operator in H, V a decomposable partial isometry such that the initial space of V(t) is equal to M a.e. t, and finally $\text{H}$(M) is the Hardy space of analytic $\text{L}$₂ vector functions with values in M ? N × N. Such operators T commute with the bilateral shift U; but, unless M = 0, T does not commute with $\text{U}{}^1$. Conversely, this is a canonical model for all R.O. with said commutativity properties. Moreover, the model is unique when T is given, and M = G(w) where w is a partial isometry in N. The detailed structure of the model is analyzed in the special case where dim N = dim M = 1. We relate the problem to a condition of Szeg? by showing that T is a $\text{R.O. iff}\text{∝}{}_0{}^{\mathrm{2\pi }}\text{log}\text{¦ V}{}_2\text{(t)¦ dt = ?∞,}\text{where}\text{V = (V}{}_1\text{, V}{}_2\text{)}$ is the partial isometry in the special case of dimension one. Szeg?'s conditions enters in a different way in the analysis of the case M = N × N, as well as in the spectral analysis of T. Our results provide an answer to a commutativity problem posed by Fuglede. If T is an arbitrary densely defined operator, and A?B(H) is normal, we prove two theorems stating conditions for the implication $\text{A}{}_{\mathrm{\cup }}\text{T ? A}{}_{\mathrm{\cup }}{}^1\text{T}$. These conditions cannot generally be relaxed.     

The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

Retraction spaces and the homotopy metric

Let X be a finite-dimensional compactum. Let $\text{R}$(X) and $\text{N}$(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2^X_h be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R^X_h be the subspace of 2^X_h consisting of the ANR's in X that are retracts of X.We show that $\text{N}$(S^m) is simply-connected for m > 1. We show that if X is an ANR and A₀?R^X_h, then lim_i→∞A_i=A₀ in 2^X_h if and only if for every retraction r₀ of X onto A₀ there are, for almost all i, retractions r_i of X onto A_i such that lim_i→∞r_i=r_o in $\text{R}$(X). We show that if X is an ANR, then the local connectedness of $\text{R}$(X) implies that of R^X_h. We prove that $\text{R}$(M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR.     

A version of Chevet's theorem for stable processes

Let $\text{X}\text{}\text{?}\text{bo}\text{Y}$ be the injective tensor product of the separable Banach spaces X and Y and let S_X, S_Y and $\text{S}{}_{\mathrm{<mtext>X\; </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>bo\; Y</mtext>}}$ be the unit spheres of these spaces. The tensor product of two symmetric finite measures η₁ on S_X and η₂ on S_Y, η₁?η₂, is defined in a natural way as a measure on $\text{S}{}_{\mathrm{<mtext>X\; </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>bo\; Y</mtext>}}$. It is shown that η₁? η₂ is the spectral measure of a p-stable random variable W on $\text{X}\text{}\text{?}\text{bo}\text{Y}$, 0 <p < 2, if and only if η₁ and η₂ are the spectral measures of p-stable random variables U and V on X and Y, respectively. Actually upper and lower bounds for $\text{(E∥ W∥}{}^{\mathrm{r}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}$ in terms of the random variables U and V are obtained. When X = C(S), Y = C(T) with S, T compact metric spaces, and η₁, and η₂ are discrete, our results imply that if θ_i, θ_ij are i.i.d. standard symmetric real valued p-stable random variables, 0 < p <2, x_i?C(S), and y_i?C(T), then the series ∑_ijθ_ijx_i(s) y_j(t) converges uniformly a.s. iff the series ∑_iθ_ix_i(s) and ∑_iθ_iy_i(t) both converge uniformly a.s. When p = 2 this follows from Chevet's theorem on Gaussian processes. Several examples are given. One of them requires an interesting upper bound on the probability distribution of the maximum of i.i.d. p-stable random variables taking values in a general Banach space.     

On σ-algebras related to the measurability of compositions

Given a measurable space (T, F), a set X, and a map ?: T → X, the σ-algebras N _Ф = ?_?∈Φ N _?, and M _Φ = ?_?∈Φ N _?, where G _?(t) = (t, ?(t)) and Φ ? X ^T , are considered. These σ-algebras are used to characterize the (F, B, ?)-measurability of the compositions g ○ ? and f о G _?, where g: X → Y, f: T × X → Y, and (Y, ?) is a measurable space. Their elements are described without using the operations ? ^?1 and G _? ^?1 .     

Sperner families over a subset

Let |X| = n > 0, |Y| = k > 0, and Y ? X. A family A of subsets of X is a Sperner family of X over Y if A₁A₂ for every pair of distinct members of A and every member of A has a nonempty intersection with Y. The maximum cardinality f(n, k) of such a family is determined in this paper. $\text{f(n,k)=(}\text{n}\text{[n}\text{2]}\text{)?(}\text{?k}\text{[n}\text{2]}\text{)}$.