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How singular are moment generating functions?  [PDF]
Eberhard Mayerhofer
Mathematics , 2011,
Abstract: This short note concerns the possible singular behaviour of moment generating functions of finite measures at the boundary of their domain of existence. We look closer at Example 7.3 in O. Barndorff-Nielsen's book "Information and Exponential Families in Statistical Theory (1978)" and elaborate on the type of exhibited singularity. Finally, another regularity problem is discussed and it is solved through tensorizing two Barndorff- Nielsen's distributions.
Applying the Moment Generating Functions to the Study of Probability Distributions  [PDF]
Informatica Economica Journal , 2007,
Abstract: In this paper, we describe a tool to aid in proving theorems about random variables, called the moment generating function, which converts problems about probabilities and expectations into problems from calculus about function values and derivates. We show how the moment generating function determinates the moments and how the moments can be used to recover the moment generating function. Using of moment generating functions to find distributions of functions of random variables is presented. A standard form of the central limit theorem is also stated and proved.
On a fixed point in the metric space of normalized Hausdorff moment sequences  [PDF]
Christian Berg,Maryam Beygmohammadi
Mathematics , 2010,
Abstract: We show that the transformation (x_n)_{n\ge 1}\to (1/(1+x_1+...+x_n))_{n\ge 1} of the compact set of sequences (x_n)_{n\ge 1} of numbers from the unit interval [0,1] has a unique fixed point, which is attractive. The fixed point turns out to be a Hausdorff moment sequence studied in papers by Berg and Dur\'an in 2008.
The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function  [PDF]
Christian Berg,Antonio J. Durán
Mathematics , 2007,
Abstract: We study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: $T((a_n))_n=1/(a_0+... +a_n)$. We determine the corresponding measure $\mu$, which has an increasing and convex density on $]0,1[$, and we study some analytic functions related to it. The Mellin transform $F$ of $\mu$ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on $]-1,\infty[$ satisfying $F(0)=1$ and the functional equation $1/F(s)=1/F(s+1)-F(s+1), s>-1$.
The converse to Curtiss' theorem for one-sided moment generating functions  [PDF]
Patrick Chareka
Mathematics , 2008,
Abstract: A set of necessary and sufficient conditions for a sequence of moment generating functions to converge to a moment generating function on an interval (a,b) not necessarily containing 0, is given. The result is derived using recent results by Mukherjea, et al. (2006) and Chareka (2007).
Perturbation of Hausdorff moment sequences, and an application to the theory of C*-algebras of real rank zero  [PDF]
George A. Elliott,Mikael Rordam
Mathematics , 2005,
Abstract: We investigate the class of unital C*-algebras that admit a unital embedding into every unital C*-algebra of real rank zero, that has no finite-dimensional quotients. We refer to a C*-algebra in this class as an initial object. We show that there are many initial objects, including for example some unital, simple, infinite-dimensional AF-algebras, the Jiang-Su algebra, and the GICAR-algebra. That the GICAR-algebra is an initial object follows from an analysis of Hausdorff moment sequences. It is shown that a dense set of Hausdorff moment sequences belong to a given dense subgroup of the real numbers, and hence that the Hausdorff moment problem can be solved (in a non-trivial way) when the moments are required to belong to an arbitrary simple dimension group (i.e., unperforated simple ordered group with the Riesz decomposition property).
Sparse Moment Sequences  [PDF]
Saroj Aryal,Farhad Jafari,Mihai Putinar
Mathematics , 2012,
Abstract: The well-known theorems of Stieltjes, Hamburger and Hausdorff establish conditions on infinite sequences of real numbers to be moment sequences. Further, works by Carath\'eodory, Schur and Nevanlinna connect moment problems to problems in function theory and functions belonging to various spaces. In many problems associated with realization of a signal or an image, data may be corrupt or missing. Reconstruction of a function from moment sequences with missing terms is an interesting problem leading to advances in image and/or signal reconstruction. It is easy to show that a subsequence of a moment sequence may not be a moment sequence. Conditions are obtained to show how rigid the space of sub-moment sequences is and necessary and sufficient conditions for a sequence to be a sub-moment sequence are established. A deep connection between the sub-moment measures and the moment measures is derived and the determinacy of the moment and sub-moment problems are related. This problem is further related to completion of positive Hankel matrices.
A non-hyponormal operator generating Stieltjes moment sequences  [PDF]
Z. J. Jablonski,I. B. Jung,J. Stochel
Mathematics , 2011,
Abstract: A linear operator $S$ in a complex Hilbert space $\hh$ for which the set $\dzn{S}$ of its $C^\infty$-vectors is dense in $\hh$ and $\{\|S^n f\|^2\}_{n=0}^\infty$ is a Stieltjes moment sequence for every $f \in \dzn{S}$ is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator $S$ which generates Stieltjes moment sequences. What is more, $\dzn{S}$ is a core of any power $S^n$ of $S$. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of indeterminate Stieltjes moment sequences. As a consequence, it is shown that there exists a non-hyponormal composition operator in an $L^2$-space (over a $\sigma$-finite measure space) which is injective, paranormal and which generates Stieltjes moment sequences. In contrast to the case of abstract Hilbert space operators, composition operators which are formally normal and which generate Stieltjes moment sequences are always subnormal (in fact normal). The independence assertion of Barry Simon's theorem which parameterizes von Neumann extensions of a closed real symmetric operator with deficiency indices $(1,1)$ is shown to be false.
An analysis of the vectorial capacity using moment-generating functions  [PDF]
Daniel A. M. Villela
Quantitative Biology , 2015,
Abstract: This paper describes a technique for analyzing the stochastic structure of the vectorial capacity using moment--generating functions. In such formulation, for an infectious disease transmitted by a vector, we obtain the generating function for the distribution of the number of infectious contacts (e.g., infectious bites from mosquitoes) between vectors and humans after a contact to a single infected individual. This approach permits us to generally derive the moments of the distribution and, under some conditions, derive the distribution function of the vectorial capacity. A stochastic modeling framework is helpful for analyzing the dynamics of disease spreading, such as performing sensitivity analysis.
Divide-and-conquer generating functions. Part I. Elementary sequences  [PDF]
Ralf Stephan
Mathematics , 2003,
Abstract: Divide-and-conquer functions satisfy equations in F(z),F(z^2),F(z^4)... Their generated sequences are mainly used in computer science, and they were analyzed pragmatically, that is, now and then a sequence was picked out for scrutiny. By giving several classes of ordinary generating functions together with recurrences, we hope to help with the analysis of many such sequences, and try to classify a part of the divide-and-conquer sequence zoo.
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