Sequence Of Probability Measures Research Articles

Tightness of Probabilities On $C(\lbrack 0, 1 \rbrack; \mathscr{Y}')$ and $D(\lbrack 0, 1 \rbrack; \mathscr{Y}')$

Let $C_\mathscr{I}' = C(\lbrack 0, 1\rbrack; \mathscr{I}')$ be the space of all continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$, where $\mathscr{L}'$ is the topological dual of the Schwartz space $\mathscr{I}$ of all rapidly decreasing functions. Let $C$ be the Banach space of all continuous functions on $\lbrack 0, 1\rbrack$. For each $\varphi \in \mathscr{I}, \Pi_\varphi$ is defined by $\Pi_\varphi:x \in C_\mathscr{I}' \rightarrow x. (\varphi) \in C$. Given a sequence of probability measures $\{P_n\}$ on $C_\mathscr{I}'$ such that for each $\varphi \in \mathscr{I}, \{P_n\Pi^{-1}_\varphi\}$ is tight in $C$, we prove that $\{P_n\}$ itself is tight in $C_\mathscr{I}'.$ A similar result is proved for the space of all right continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$.

A note on contiguity and L1-norm

The interrelationship between the property of contiguity of two sequences of probability measures { P n } and { Q n } and the convergence to zero of ∥ P n − Q n ∥ is extendended and clarified.

Gleichverteilung auf Diskreten Halbgruppen

Let (μn) be a sequence of probability measures on a discrete semigroup S. In this paper we study the (left, right) uniform distribution of (μn), i.e. the vague or norm convergence of μn−δxμn→0 for every x in S δx means the point measure of x and δxμn the convolution of δx and μn). We shall give equivalent conditions for the (left, right) uniform distributed sequences and relate the existence of such sequences to the (left, right) amenable semigroups. In the last section we consider especially the uniform distribution of the convolution sequence μn of a probability measure.

More on limit theorems for iterates of probability measures on semigroups and groups

In this paper, we continue earlier works of one of the authors on vague convergence of the sequence βk,n=βk+1 *...*βn, where βn is a sequence of probability measures on semigroups or groups. Typical results in this paper are: Theorem. Let S be a locally compact noncompact second countable group such that \(S = \overline {\bigcup\limits_{n = 1}^\infty {S_\beta ^n ,} } S_\beta\) being the support of a probability measure β on S. Suppose there exists an open set V with compact closure such that x−1Vx=V for every x∈S. Then for all compact sets K, sup{βn(Kx): x∈S→0 as n→∞. Theorem. Let S be an at most countable discrete group. Let βn be a sequence of probability measures on S. Then for all nonnegative integers k, the sequence βk,n converges vaguely to some probability measure if and only if there exists a finite subgroup G such that the series \(\sum\limits_{n = 1}^\infty {\beta _n } (S - G) s>0 for all n, e being the identity.

A local limit theorem for the convolution of probability measures on a compact connected group

Given a sequence of probability measures on a compact connected group we give an estimate on the speed of convergence to 1 of the density function of their n-fold convolution. Our method is an adaptation of the characteristic function technique to this case.

Weak convergence of probability measures relative to incompatible topology and ?-field with applications to renewal theory

Necessary and sufficient conditions are given in order that a sequence of probability measures, weakly convergent relative to a given topology τ 0 and associated σ-field σ(τ 0), are weakly convergent (and satisfy a continuity theorem) relative to the σ(τ 0)-measurable functions which are continuous in some finer topology τ 1, even if μ does not extend to σ(τ 0). These conditions are shown to be applicable to a sequence of translated renewal measures. Alternate conditions (tightness, uniformity of weak convergence) are investigated and shown to be inappropriate.

Convergence of recursive adaptive and identification procedures via weak convergence theory

Results and concepts in the theory of weak convergence of a sequence of probability measures are applied to convergence problems for a variety of recursive adaptive (stochastic approximation-like) methods. Similar techniques have had wide applicability in areas of operations research and in some other areas in stochastic control. It is quite likely that they will play a much more important role in control theory than they do at present, since they allow relatively simple and natural proofs for many types of convergence and approximation problems. Part of the aim of the paper is tutorial: to introduce the ideas and to show how they might be applied. Also, many of the results are new, and they can all be generalized in many directions.

On Ergodic Sequences of Measures

Let $Z$ be the group of integers and $\bar Z$ its Bohr compactification. A sequence of probability measures $\{ {\mu _n},n = 1,2, \ldots \}$ defined on $Z$ is said to be ergodic provided ${\mu _n}$ converges weakly to $\bar \mu$, the Haar measure on $\bar Z$. Let ${A_n} \subset Z,n = 1,2, \ldots$ and define ${\mu _n}$ by ${\mu _n}(B) = |{A_n} \cap B|/|{A_n}|$ where $|B|$ is the cardinality of $B$. Then it is easy to show that if $|{A_n} \cap {A_n} + k|/|{A_n}| \to 1$ for every $k \in Z$, then ${\mu _n}$ is ergodic. Let $0 \leq {p_k} \leq 1$. In this paper we construct (random) sequences $\{ {\mu _n}\}$ which are ergodic, and such that $\lim (|{A_n} \cap {A_n} + k|/|{A_n}|) = {p_k}$, for every $k \in Z$.

Necessary conditions for the convergence of kullback-leibler’s mean information

Two types of necessary conditions are given for the convergence of Kullback-Leibler’s mean information, one of which is connected with an asymptotic equivalence of two sequences of probability measures, and in special cases, with convergence of a sequence of probability distributions. The other is given in terms of the generalized probability density functions.