Infinite-dimensional Vector Space Research Articles

Measures on infinite dimensional vector spaces

Measures on infinite dimensional vector spaces

Central Limit Theorems for Families of Sequences of Random Variables

Let $F$ be a (nonempty) set of distribution functions (d.f.'s) of random variables (r.v.'s) with zero means and positive, finite variances. Let $\mathfrak{F}(F)$ be the set of all sequences of independent r.v.'s (independent within each sequence) whose d.f.'s belong to $F$ but are not necessarily the same from term to term of the sequence. No assumptions are made on the interrelations between the joint probability spaces of the r.v.'s of different sequences of $\mathfrak{F}(F)$. Accordingly, dependence or independence between r.v.'s of different sequences needs not be specified. A generic member of $\mathfrak{F}(F)$ will be denoted by $\epsilon = \{\epsilon_k; k = 1, 2, \cdots\}$, or, when we discuss sequences of members of $\mathfrak{F}(F)$, by $\epsilon(n) = \{\epsilon_{nk}; k = 1, 2, \cdots\}, n = 1, 2, \cdots$. In the following, $\mathfrak{F}(F)$ plays the role of a parameter space, the parameter points being $\epsilon, \epsilon(1), \epsilon(2), \cdots$. Since only the d.f.'s of a sequence of (independent) r.v.'s are relevant to the central limit theorem, instead of $\mathfrak{F}(F)$, the set of sequences of d.f.'s corresponding to the elements of $\mathfrak{F}(F)$ may also be regarded as the parameter space. It clearly is a map of $\mathfrak{F}(F)$. The inverse mapping subdivides $\mathfrak{F}(F)$ into certain equivalence classes. From elementary set theory it follows that $\mathfrak{F}(F)$ as well as its map has the cardinality of the continuum if $F$ contains more than one element. Further, let $\{a_{nk}; n = 1, 2, \cdots; k = 1, 2, \cdots\}$ be a double sequence of real constants, and let $\{k_n\}$ be a sequence of positive integers such that $a_{nk_n} \neq 0$ and $a_{nk} = 0$ for $k > k_n, n = 1, 2, \cdots$. We denote the variance of $\sum^{k_n}_{k=1} a_{nk}\epsilon_{nk}$ by \begin{equation*}\tag{1}B_n^2 = \sum^{k_n}_{k=1} a^2_{nk}\sigma^2_{nk}, \quad\sigma^2_{nk} = \operatorname{var} \epsilon_{nk},\end{equation*} and put \begin{equation*}\tag{2}\zeta_n = B_n^{-1} \sum^{k_n}_{k=1} a_{nk}\epsilon_{nk},\quad n = 1, 2, \cdots,\end{equation*} where $B_n = + (B_n^2)^{\frac{1}{2}}$. If all sequences $\epsilon(n)$ are identical we denote them by $\epsilon$ and write instead of (2) $\zeta_n(\epsilon) = B_n^{-1}(\epsilon) \sum^{k_n}_{k=1} a_{nk}\epsilon_k, \quad n = 1, 2, \cdots,$ which shows the dependence on the parameter $\epsilon$ more clearly; $B_n^2(\epsilon)$ is defined analogously to (1). This note deals with necessary and sufficient conditions on the set $F$ and on the double sequence $\{a_{nk}\}$ in order that the d.f.'s of the $\zeta_n$ tend to the standard normal d.f. (denoted by $\Phi(x)$) and that the $B^{-1}_na_{nk}\epsilon_{nk}$ are infinitesimal for every sequence of sequences $\epsilon(1), \epsilon(2), \cdots \varepsilon \mathfrak{F}(F)$ (Theorem 1). For the case of identical sequences $\epsilon(n) (= \epsilon)$, which is of particular interest in applications, a normal convergence theorem (Theorem 3) holding uniformly for $\epsilon$ on $\mathfrak{F}(F)$ is obtained under the same necessary and sufficient conditions as in Theorem 1. Theorem 3 yields, e.g., conditions for the asymptotic normality of the least squares estimators of the parameters in a linear regression with independent and not necessarily identically distributed errors whose d.f.'s are unknown but belong to a certain class $F$ (Eicker (1963)). From the necessity of the conditions it follows that these conditions are the best possible ones under the limited information about the error terms provided by the model assumptions. Frequently, when limit theorems for families of sequences of random variables are met in statistics and probability theory, the emphasis is on the uniformity of the convergence of the sequences with respect to the family parameter which assumes values in a given a priori set. (For an example, compare Parzen (1954), p. 38. That paper also cites some of the earlier publications on the subject.) The present note emphasizes not primarily the uniformity of the convergence, but the necessity of the conditions (including one on the parameter space $\mathfrak{F}(F))$ for the convergence on the parameter space. Accordingly, the set $F$ cannot be prescribed arbitrarily. The uniformity of the convergence on $\mathfrak{F}(F)$ is shown without difficulty to be implied by the ordinary convergence on $\mathfrak{F}(F)$. Another difference from earlier work may be seen in the particular structure of $\mathfrak{F}(F)$, which is comparable to an infinite-dimensional vector space. Although many of the earlier convergence theorems are (or can be) formulated for abstract parameter spaces, in the applications such as estimation theory these spaces are usually specialized to intervals on the line or in a finite-dimensional vector space.

On a special group of isometries of an infinite dimensional vectorspace

On a special group of isometries of an infinite dimensional vectorspace

On Schauder Bases for Spaces of Continuous Functions1)

In a finite dimensional vector space V a set xi, i = 1, 2, …, n of vectors of V is said to be a basis, base, or coordinate system for V if the vectors xi are linearly independent and if each vector in V is a linear combination of the elements x1 with real coefficients. If a topology for V is defined in terms of a norm ||.|| then {xi} is a basis for V if and only if to each x ϵ V corresponds a unique set of constants ai such thatIn infinite dimensional normed vector spaces the above concepts of basis have different generalizations. The first or algebraic definition gives a Hamel basis which is a maximal linearly independent set [l, p. 2]. We shall be interested in the other or topological definition.

The Automorphism Group of a Lie Group

Introduction. The group A (G) of all continuous and open automorphisms of a locally compact topological group G may be regarded as a topological group, the topology being defined in the usual fashion from the compact and the open subsets of G (see ?1). In general, this topological structure of A (G) is somewhat pathological. For instance, if G is the discretely topologized additive group of an infinite-dimensional vector space over an arbitrary field, then A (G) already fails to be locally compact. On the other hand, if G is a connected Lie group, we shall show without any difficulty that the compact-open topology of A (G) coincides with the topology obtained by identifying A (G) with a closed subgroup of the linear group of automorphisms of the Lie algebra of G, as was done by Chevalley (in [1]) in order to make A (G) into a Lie group. We shall then deduce that A (G) is a Lie group whenever the group of its components, G/Go, is finitely generated('), where Go denotes the component of the identity element in G. The other questions with which we shall be concerned are the following: Let I(Go) denote the group of the inner automorphisms of Go, and let E(Go, G) denote the natural image in A (Go) of A (G). Regard I(Go) and E(Go, G) as

A Topology for the Set of Primitive Ideals in an Arbitrary Ring

1. In a recent paper we have called a ring ? primitive if a contains a maximal right ideal a whose quotient 3: P2 = 0.1 In general if a is any right ideal 3: W is the totality of elements b such that xb e S for all x in 1. 3: 2t is a two-sided deal and if 21 has an identity, : W1 is the largest twosided ideal of W1 contained in '. The primitive rings appear to play the same r61le in the general structure theory of rings that is played by simple rings in the classical theory of rings that satisfy the descending chain condition for one-sided ideals. Corresponding to the Wedderburn-Artin structure theorem on simple rings satisfying the descending chain condition we have the theorem that if 21 is a primitive ring 7 0, W1 is isomorphic to a dense ring of linear transformations in a suitable vector space over a division ring. W e shall call a two-sided ideal e3 in W1 a primitive ideal if Wg ^ 21 and 21 !3 is a primitive ring. It is known that if 82 is not a radical ring then 21 contains primitive ideals. Moreover, in this case the intersection II3 of all the primitive ideals in 2t coincides with the radical T of 91. In particular if 1 is semi-simple, II3 = 0. If 21 is a ring with an identity, 21 is not a radical ring. Hence if (S is any two-sided ideal 7 21 in W1, W1 contains a primitive ideal 3 6. Since W2 e3 (21 S) (3 (), e3 is primitive in 21. Thus if 21 is a ring with an identity any two-sided ideal E # 1 can be imbedded in a primitive ideal. Any commutative primitive ring is a field. Consequently any primitive ideal in a commutative ring is maximal. On the other hand, in a noncommutative ring there may exist primitive ideals that are not maximal. For example, the ring 8 of all linear transformations in an infinite dimensional vector space 9a over a division ring is primitive. Hence (0) is a primitive ideal. However, 8 contains as a proper two-sided ideal the set a of finite valued linear transformations in T9.