Homogeneous Random Process Research Articles

Power series whose coefficients form homogeneous random processes

Power series f(z) =⌆ aizi are considered, where the sequence {ai} forms a homogeneous random process. If the sequence is exchangeable and the variance of the marginal distributions exists, it is proved that r, the random radius of convergence of f(z), takes the values 0 and 1. If the sequence is a second order stationary time series then r=1 with probability 1. If {ai} is a regular denumerable Markov chain, it can be proved that r=c≲=1 with probability 1, but both c=0 and c=1 can arise. A number of criteria are given for deciding the value of c in this situation.

On the probability function of a homogeneous random process

Abstract d1. In his famous paper [4] Kohnogoroff has stated—under very general conditions—the general form of the characteristic function (c.f.) of a homogeneous random process or, as it is today mostly called, a process with independent and stationary increments.

On the Existence of Exact Upper Sequences

Previous article Next article On the Existence of Exact Upper SequencesB. A. RogozinB. A. Rogozinhttps://doi.org/10.1137/1113081PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. Strassen, A converse to the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 4 (1965/1966), 265–268 10.1007/BF00539114 MR0200965 0141.16501 CrossrefGoogle Scholar[2] A. IA. Khinchin, Two theorems on stochastic processes with monotypic increments, Mat. Sb., 4 (1938), 577–584, (In Russian.) Google Scholar[3] B. V. Gnedenko, On the growth of homogeneous random processes with independent single-type increments, C. R. (Doklady) Acad. Sci. URSS (N. S.), 40 (1943), 90–93, (In Russian.) MR0011400 0063.01647 Google Scholar[4] Paul Lévy, Loi faible et loi forte des grands nombres, Bull. Sci. Math. (2), 77 (1953), 9–40 MR0054878 0053.40903 Google Scholar[5] P. Lévy, Theorie de l'Addition des Variables Aleatoires, Paris, 1937 Google Scholar[6] Michel Loève, Probability theory. Foundations. Random sequences, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955xv+515 MR0066573 0066.10903 Google Scholar[7] Kai Lai Chung, On the renewal theorem in higher dimensions, Skand. Aktuarietidskr., 35 (1952), 188–194 (1953) MR0054879 0048.11103 Google Scholar[8] W. Feller, A limit theorem for random variables with infinite moments, Amer. J. Math., 68 (1946), 257–262 MR0016569 0060.28704 CrossrefGoogle Scholar[9] Carl-Gustav Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math., 77 (1945), 1–125 MR0014626 0060.28705 CrossrefGoogle Scholar[10] L. D. Meshalkin and , B. A. Rogozin, An estimate of the distance between distribution functions based on the closeness of their characteristic functions and its application to the central limit theorem, in “Limit Theorems”, Tashkent, (1963), 49–56, (In Russian.) Google Scholar[11] Horst Oswald, Über eine Abschätzung des Restgliedes im zentralen Grenzverteilungssatz, Wiss. Z. Friedrich-Schiller-Univ. Jena/Thüringen, 14 (1965), 261–268 MR0236975 0151.22603 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Chris Heyde’s Work in Probability Theory, with an Emphasis on the LILSelected Works of C.C. Heyde | 24 June 2010 Cross Ref Sample Functions of the Gaussian ProcessSelected Works of R.M. Dudley | 2 July 2010 Cross Ref Some effects of trimming on the law of the iterated logarithmJournal of Applied Probability, Vol. 41, No. A | 14 July 2016 Cross Ref Some effects of trimming on the law of the iterated logarithmJournal of Applied Probability, Vol. 41, No. A | 14 July 2016 Cross Ref Limit theorems for mixing sequences without rate assumptionsThe Annals of Probability, Vol. 26, No. 2 | 1 Apr 1998 Cross Ref A universal form of the Chung-type law of the iterated logarithmThe Annals of Probability, Vol. 25, No. 4 | 1 Oct 1997 Cross Ref On strong convergence of arraysBulletin of the Australian Mathematical Society, Vol. 50, No. 2 | 17 April 2009 Cross Ref On the Monotonicity of Exact Upper Sequences for Sums of Independent Random Vectors in Hilbert SpaceTheory of Probability & Its Applications, Vol. 33, No. 1 | 17 July 2006AbstractPDF (330 KB)On the law of the iterated logarithm for nonidentically distributed random variablesStochastic Analysis and Applications, Vol. 6, No. 2 | 1 Jan 1988 Cross Ref A functional law of the iterated logarithm for distributions in the domain of partial attraction of the normal distributionStochastic Processes and their Applications, Vol. 27 | 1 Jan 1987 Cross Ref Regular Methods of Summing Random TermsTheory of Probability & Its Applications, Vol. 30, No. 1 | 28 July 2006AbstractPDF (779 KB)On the Existence of Exact Upper SequencesTheory of Probability & Its Applications, Vol. 29, No. 2 | 17 July 2006AbstractPDF (315 KB)On Lévy–Kolmogorov Inequalities for Banach-Space-Valued Random VariablesTheory of Probability & Its Applications, Vol. 29, No. 4 | 17 July 2006AbstractPDF (527 KB)Criteria for Strong Convergence of Normalized Sums of Independent Random Variables and Their ApplicationsTheory of Probability & Its Applications, Vol. 29, No. 3 | 17 July 2006AbstractPDF (1300 KB)One-Sided Versions of Strong Limit TheoremsTheory of Probability & Its Applications, Vol. 28, No. 1 | 17 July 2006AbstractPDF (1252 KB)An extension of kesten's generalised law of the iterated LogarithmBulletin of the Australian Mathematical Society, Vol. 21, No. 3 | 17 April 2009 Cross Ref On one-sided boundedness of normed partial sumsBulletin of the Australian Mathematical Society, Vol. 21, No. 3 | 17 April 2009 Cross Ref A remark on the log-log lawZeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 26, No. 2 | 1 Jan 1973 Cross Ref Volume 13, Issue 4| 1968Theory of Probability & Its Applications History Submitted:23 May 1967Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1113081Article page range:pp. 667-672ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Dams in series with continuous release

In a previous paper (Moran (1956)) the theory of a dam with a continuous release was developed for a situation in which the input was defined by an additive homogeneous random process of gamma distribution type, and the release was defined to occur continuously at a fixed rate when the content of the dam was non-zero. In the present paper we modify these conditions by assuming first that the input is defined by a general additive homogeneous process with non-negative increments with finite second moment, and secondly that the rate of release is proportional to the content of the dam. This modification to the release rule ensures that the content of the dam is never zero and the theory is then so simplified that the distributional properties of the contents of a sequence of dams in series can be easily found.

Dams in series with continuous release

In a previous paper (Moran (1956)) the theory of a dam with a continuous release was developed for a situation in which the input was defined by an additive homogeneous random process of gamma distribution type, and the release was defined to occur continuously at a fixed rate when the content of the dam was non-zero. In the present paper we modify these conditions by assuming first that the input is defined by a general additive homogeneous process with non-negative increments with finite second moment, and secondly that the rate of release is proportional to the content of the dam. This modification to the release rule ensures that the content of the dam is never zero and the theory is then so simplified that the distributional properties of the contents of a sequence of dams in series can be easily found.

Comparison of Time‐Delay Spread and Spatial Coherence for Ionospheric Propagation

This paper describes spatial selective fading (interference patterns due to modal arrival angle differences) as a homogeneous random process. It is shown that for ionospheric propagation, the power spectrum in vector wave number (wave number dispersion profile) should be of approximately the same shape as the power spectrum in time delay (time‐delay profile). On the basis of this similarity, it is suggested that the correlation between signals from spaced antennas (spatial coherence) should be inversely proportional to rms time‐delay spread.An experiment was performed on a 4100‐km path at 7 and 14 Mc/s to verify this assumption. Results are generally in agreement with theory, and discrepancies that do exist are analyzed.

Scattering of plane waves by locally homogeneous dielectric noise

The scattering of acoustic and electromagnetic radiation by random refractive index fluctuations of the scattering medium (e.g. the troposphere, the ocean) is a problem of considerable contemporary interest. For brevity, we shall refer to these refractive index fluctuations as dielectric noise. The purpose of this paper is to compare two approaches to the problem of the scattering of incident plane waves by a confined region of dielectric noise. In the first approach, which is the conventional one, the scattering dielectric noise is represented as a ‘section’ of a homogeneous (i.e. spatially stationary) random process. In the second approach, which is based on a recent generalization of the notion of stationarity (1), the scattering dielectric noise is represented as a locally homogeneous random process. The advantage of representing the dielectric noise as a locally homogeneous process, rather than as a homogeneous process, is that one can thereby take into account in a natural way the spatial variation of the average noise power, without departing from the physically reasonable requirement that the process be approximately homogeneous in suitably restricted domains.

On a Characterization of Brownian Motion

Let $X(t)$ be a homogeneous random process with independent increments, and $a(t)$, $b(t)$ certain functions that are continuous in the closed interval a, b. Then the following theorem is true:If\[ \int_a^b {[a(t)b(t)]^2 dt \ne 0,} \] the integrals\[ Y = \int_a^b {a(t)dX(t)} {\textit{ and }}Z\int_a^b {b(t)dX(t)} \]are independent, and if at least one of the integrals\[ \int_a^b {\frac{{a^2 (t)}} {{b^2 (t)}}dt\,} ,\int_a^b {\frac{{b^2 (t)}} {{a^2 (t)}}dt\,} \]exists, then the process$X(t)$is a Brownian motion.A similar theorem is proved for infinite linear forms\[ \sum\limits_{k = 1}^\infty {a_k X_k } ,\sum\limits_{k = 1}^\infty {b_k X_k } .\]

On the mathematical theory of risk

Although the first essay on the theory of risk (41) was published in 1786, this theory has been for the most part developed during this century. Moreover, the original presentation was rather difficult and the formulae inconvenient for the practical computer. Only recently has the classical theory of risk attained a satisfactory and seemingly final form. Few text-books, none of them in the English language, discuss these developments; it may therefore be useful to give an exposition of these results. As a rule no proofs will be given, instead reference will be made to the relevant literature.This survey will be restricted to the classical theory of risk, which deals essentially with the individual insurance. It is possible to approach the problem of risk from an entirely different angle, considering not the individual insurance but all the policies in force. This leads to the collective theory of risk (27, 29, 15, 40), which is based upon the theory of homogeneous random processes. Only the classical theory will be considered here, the inclusion of the collective theory of risk would considerably extend the scope of this paper, and in any case many of the relevant papers are in English.

A Special Class of Homogeneous Random Processes

A Special Class of Homogeneous Random Processes