Extreme nth moments for distributions on [0, 1] and the inverse of a moment space map

1968 ◽  
Vol 5 (3) ◽  
pp. 693-701 ◽  
Author(s):  
Morris Skibinsky
Keyword(s):  
Positive Integer ◽  
Probability Measure ◽  
Unit Interval ◽  
Image Position

Let n be a positive integer. Denote by Mn the convex, closed, bounded, and n-dimensional set of all n-tuples (c1, c2,···, cn) such that for some probability measure a on the Borel subsets of the unit interval I = [0,1].

Extreme nth moments for distributions on [0, 1] and the inverse of a moment space map

1968 ◽  
Vol 5 (03) ◽  
pp. 693-701 ◽  
Author(s):  
Morris Skibinsky
Keyword(s):  
Positive Integer ◽  
Probability Measure ◽  
Unit Interval ◽  
Image Position

Let n be a positive integer. Denote by Mn the convex, closed, bounded, and n-dimensional set of all n-tuples (c 1, c 2,···, c n) such that for some probability measure a on the Borel subsets of the unit interval I = [0,1].

scholarly journals The essential norms of composition operators

1992 ◽  
Vol 34 (2) ◽  
pp. 143-155 ◽  
Author(s):  
Boo Rim Choe
Keyword(s):  
Positive Integer ◽  
Hardy Space ◽  
Probability Measure ◽  
Composition Operators ◽  
Invariant Probability Measure ◽  
Open Unit ◽  
Rotation Invariant ◽  
Open Unit Ball ◽  
Image Position ◽  
Fixed Positive Integer

Throughout the paper n denotes a fixed positive integer unless otherwise specified. Let B = Bn denote the open unit ball of ℂn and let S = Sn denote its boundary, the unit sphere. The unique rotation-invariant probability measure on 5 will be denoted by σ = σn. For n = l, we use more customary notations D = B1, T = S1 and dσ1= dθ/2π. The Hardy space on B, denoted by H2(B), is then the space of functions f holomorphic on B for which

The range of the (n + 1)th moment for distributions on [0, 1]

1967 ◽  
Vol 4 (03) ◽  
pp. 543-552 ◽  
Author(s):  
Morris Skibinsky
Keyword(s):  
Positive Integer ◽  
Convex Hull ◽  
Higher Order ◽  
Unit Interval ◽  
Space Curve ◽  
Order Moment ◽  
Probability Measures ◽  
Image Position ◽  
Moment Spaces

Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn ): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn ) of Mn , define Note that v −, v + depend only on C1, C2, …, Cn− 1; Vm only on cn ; We shall as notational convenience dictates and as will be apparent from the context regard v ± n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.

The range of the (n + 1)th moment for distributions on [0, 1]

1967 ◽  
Vol 4 (3) ◽  
pp. 543-552 ◽  
Author(s):  
Morris Skibinsky
Keyword(s):  
Positive Integer ◽  
Convex Hull ◽  
Higher Order ◽  
Unit Interval ◽  
Space Curve ◽  
Order Moment ◽  
Probability Measures ◽  
Image Position ◽  
Moment Spaces

Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn) of Mn, define Note that v−, v+ depend only on C1, C2, …, Cn− 1; Vm only on cn; We shall as notational convenience dictates and as will be apparent from the context regard v±n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

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