Dependent Random Vectors Research Articles

Limit theorems for random sums of dependent d-dimensional random vectors

Let there be given a doubly infinite sequence ~ = { { X , k ; k e N } n e N } of ddimensional random vectors (rv's) defined on a common probability space 4 " (f2,~,,P) and a doubly infinite sequence 5 = { { ,k, ke{0}wN};neN} of o-subfields of ~, which is row-wise increasing ~ a ~1) and adapted to \~ n,k ~ n,k+ 9 , i.e. every X,k is ~,k-measurable. A pair (~, 5) will be called a system. Moreover, let us choose a fixed sequence of random variables vm: (f2, o ~, P)-~ N u { + o o } , meN, such that every v m is a stopping time with respect to the m-th row of ~, i.e. [vmNk]e~m, 1, k e N . Finally, let us define a sequence of random sums: v,

On the rate of approximation in the central limit theorem for dependent random variables and random vectors

Large “ O” and small “ o” approximations of the expected value of a class of smooth functions ( f ∈ C r ( R)) of the normalized partial sums of dependent random variable by the expectation of the corresponding functions of normal random variables have been established. The same types of approximations are also obtained for dependent random vectors. The technique used is the Lindberg-Levy method generalized by Dvoretzky to dependent random variables.

A Central Limit Theorem for Spatial Samples

Geographical AnalysisVolume 12, Issue 4 p. 299-324 Free Access A Central Limit Theorem for Spatial Samples Tony E. Smith, Tony E. Smith This research was supported by the National Science Foundation under Grant SES 79–19457. The author is indebted to Jörgen W. Weibull and to the members of the Regional Theory Workshop for helpful comments on an earlier draft of this paper. Tony E. Smith is professor of regional science, University of Pennsylvania.Search for more papers by this author Tony E. Smith, Tony E. Smith This research was supported by the National Science Foundation under Grant SES 79–19457. The author is indebted to Jörgen W. Weibull and to the members of the Regional Theory Workshop for helpful comments on an earlier draft of this paper. Tony E. Smith is professor of regional science, University of Pennsylvania.Search for more papers by this author First published: October 1980 https://doi.org/10.1111/j.1538-4632.1980.tb00039.xCitations: 10 This research was supported by the National Science Foundation under Grant SES 79–19457. The author is indebted to Jörgen W. Weibull and to the members of the Regional Theory Workshop for helpful comments on an earlier draft of this paper. Tony E. Smith is professor of regional science, University of Pennsylvania. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat LITERATURE CITED Bhattacharaya, R. N., and R. R. Rao (1976). Normal Approximation and Asymptotic Expansions, New York: Wiley. Billingsley, P. (1979). Probability and Measure, New York: Wiley. Bulinskii, A. V. and I. G. Zhurbenko (1976). “A Central Limit Theorem for Additive Random Functions,” Theory of Probability and its Applications, 21, 687– 97. Cliff, A. D., and J. K. Ord (1973). Spatial Autocorrelation, London: Pion. Cochran, W. G. (1953). Sampling Techniques, New York: Wiley. Diananda, P. H. (1955). “The Central Limit Problem for m-Dependent Variables,” Proceedings of the Cambridge Philosophical Society, 51, 92– 95. Dugundji, J. (1966). Topology, Boston: Allyn and Bacon. Dvoretsky, A. (1972). “Asymptotic Normality of Sums of Dependent Random Variables,” Proceedings of the Sixth Berkeley Symposium, 2, 513– 34. Dvoretsky, A. (1977). “Asymptotic Normality of Sums of Dependent Random Vectors,” in Multivariate Analysis-IV, edited by P. R. Krishnaiah, 23– 34. Haining, R. P. (1978). “The Moving Average Model for Spatial Interaction,” Transactions of the Institute of British Geographers, 3, 202– 5. Haining, R. P. (1979). “Statistical Tests and Process Generators for Random Fields,” Geographical Analysis, 11, 45– 64. Halmos, P. (1950). Measure Theory, New York: Van Nostrand Reinhold. Hardy, G. H., J. E. Littlewood, and G. Pólya (1934). Inequalities, Cambridge, England: Cambridge University Press. Hoeffding, W., and H. Robbins (1948). “The Central Limit Theorem for Dependent Variables,” Duke Mathematical Journal, 15, 773– 80. Petrov, V. V. (1957). Sums of Independent Random Variables, Berlin: Springer-Verlag. Rao, C. R. (1973). Linear Statistical Inference and its Applications, New York: Wiley. Rozanov, Yu. A. (1960). “A Central Limit Theorem for Additive Random Functions,” Theory of Probability and its Applications, 5, 221– 23. Rosenblatt, M. (1956). “A Central Limit Theorem and Mixing Condition,” Proceedings of the National Academy of Sciences, 42, 43– 47. Rosenblatt, M. (1971). Markov Processes: Structure and Asymptotic Behavior, Berlin: Springer-Verlag. Sen, A. K. (1976). “Large Sample-Size Distribution of Statistics used for Testing Spatial Autocorrelation,” Geographical Analysis, 9, 175– 84. Sen, A. K., and S. Sööt (1977). “Rank Tests for Spatial Correlation,” Environment and Planning A, 9, 897– 903. Simmons, G. F. (1963). Topology and Modern Analysis, New York: McGraw-Hill. Tobler, W. (1975). “Linear Operators Applied to Areal Data,” in Display and Analysis of Spatial Data, edited by J. C. Davis and M. J. McCullagh, pp. 14– 37. New York: Wiley. Tobler, W. (1979). “Lattice Tuning,” Geographical Analysis, 11, 36– 44. Volkonskii, V. A. and Yu. A. Rozanov (1959). “Some Limit Theorems for Random Functions,” Theory of Probability and its Applications, 4, 178– 97. Whittle, P. (1954). “On Stationary Processes in the Plane,” Biometrika, 41, 434– 39. Wilks, S. S. (1962). Mathematical Statistics, New York: Wiley. Citing Literature Volume12, Issue4October 1980Pages 299-324 ReferencesRelatedInformation

A remark on the conditioning in limit theorems for dependent random vector in $R^d$

A remark on the conditioning in limit theorems for dependent random vector in $R^d$

Approximation Thorems for Independent and Weakly Dependent Random Vectors

In this paper we prove approximation theorems of the following type. Let $\{X_k, k \geqslant 1\}$ be a sequence of random variables with values in $\mathbb{R}^{d_k}, d_k \geqslant 1$ and let $\{G_k, k \geqslant 1\}$ be a sequence of probability distributions on $\mathbb{R}^{d_k}$ with characteristic functions $g_k$ respectively. If for each $k \geqslant 1$ the conditional characteristic function of $X_k$ given $X_1, \cdots, X_{k - 1}$ is close to $g_k$ and if $G_k$ has small tails, then there exists a sequence of independent random variables $Y_k$ with distribution $G_k$ such that $|X_k - Y_k|$ is small with large probability. As an application we prove almost sure invariance principles for sums of independent identically distributed random variables with values in $\mathbb{R}^d$ and for sums of $\phi$-mixing random variables with a logarithmic mixing rate.

Central Limit Theorems for Sums of Dependent Vector Variables

We prove the following central limit theorems for sums of mutually dependent random vector variables: Given that a sequence of random vector variables satisfies a certain type of decoupling condition (and two milder restrictions), we present a Lindeberg-Feller condition which we show to be both necessary and sufficient for central limit behavior. The decoupling condition and one of the two milder conditions is then applied to a Markov process with stationary transition mechanism.

Convergence of one partial form of sums of dependent random vectors to a normal distribution

The method of moments is used to establish an asymptotic normal distribution for a sequence of dependent random vectors of special form.