Donsker's Invariance Principle Research Articles

Large deviations and stochastic homogenization

A general theorem is stated providing large deviations estimates for a family of measures on a topological vector space. Applications are given in the second part, where large deviations problems arising in stochastic homogenization are discussed. Another application is given in similar problems connected with Donsker's invariance principle.

The central limit theorem for the lorentz gas and martingales

The original proof of the Central limit theorem for Lorentz gas was given by L. A.Bunimovich and Ya. G. Sinai in 1981. In this paper a much more concise proof using Donsker's invariance principle for martingales is given.

Statistical properties of lorentz gas with periodic configuration of scatterers

In our previous paper Markov partitions for some classes of dispersed billiards were constructed. Using these partitions we estimate the decay of velocity auto-correlation function and prove the central limit theorem of probability theory and Donsker's Invariance Principle for Lorentz Gas with periodic configuration of scatterers.

An invariance principle in k-dimensional extended renewal theory

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ ) v, τ= 1,· ··, k is the covariance matrix of In this paper the weak convergence of Z n in (D[0, ∞)) k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t 1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ 1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.

An invariance principle in k-dimensional extended renewal theory

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ)v, τ= 1,· ··,k is the covariance matrix of In this paper the weak convergence of Zn in (D[0, ∞))k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.

On the invariance principle for signed-rank statistics

continuous a.e. [#], the limit here being in the sense of weak convergence of distributions. Lamperti [5] improved Donsker's result in the following way: he showed that (1) holds for a much larger class of functionals, provided further assumptions are made on the moments of the Xi's. This was achieved by replacing the space C[0, 1] of continuous functions by the space Lip, of functions which are H61der continuous of order c~. If the Xi's have finite moments of all orders, then Corollary 1 of Lamperti [5] says that (1) holds for any functional which is continuous in the topology of Lips, a.e. [/~] on Lip~, for some 7 89 A result analogous to Donsker's invariance principle was proved by Sen [8] for signed-rank statistics. In its general form Sen's theorem is false (Sen [9]). However Sen's proof is correct for the statistics considered here.

A quantum-mechanical functional central limit theorem

Continuing an earlier work [4], properties of canonical Wiener processes are investigated. An analog of the sample path continuity property is obtained. A noncommutative counterpart of weak convergence is formulated. Operator processes ( P n , Q n ) analogous to the random-walk approximating processes of the Donsker invariance principle are defined in terms of a sequence ( p i , q i ) of pairs of quantum mechanical canonical observables satisfying hypotheses analogous to those of the classical central limit theorem. It is shown that P n , Q n ) converges weakly to a canonical Wiener process.

On the Rate of Convergence of Distributions of Functionals of Integral Type

Previous article Next article On the Rate of Convergence of Distributions of Functionals of Integral TypeI. S. BorisovI. S. Borisovhttps://doi.org/10.1137/1121034PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Stanley Sawyer, Rates of convergence for some functionals in probability, Ann. Math. Statist., 43 (1972), 273–284 MR0301782 0239.60024 CrossrefGoogle Scholar[2] A. V. Skorokhod, Investigations in the Theory of Random Processes, Izd. Kievsk. Un-ta, (1961), , (In Russian.) Google Scholar[3] A. A. Borovkov, Notes on inequalities for sums of independent random variables, Theory Prob. Applications, 17 (1972), 556–557 10.1137/1117068 0283.60054 LinkGoogle Scholar[4] S. V. Nagaev and , D. Kh. Fuk, Probability inequalities for sums of independent random variables, Theory Prob. Applications, 16 (1971), 643–660 10.1137/1116071 0259.60024 LinkGoogle Scholar[5] V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York, 1975x+346 MR0388499 0322.60042 CrossrefGoogle Scholar[6] Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963xvi+685 MR0203748 Google Scholar[7] D. H. Fuk, Certain probabilistic inequalities for martingales, Sibirsk. Mat. Ž., 14 (1973), 185–193, 239, (In Russian.) MR0326835 Google Scholar[8] A. A. Borovkov, On the rate of convergence for the invariance principle, Theory Prob. Applications, 18 (1973), 207–225 10.1137/1118025 0323.60031 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails A Refinement of Asymptotics in the Prokhorov–Donsker Invariance Principle for Integral FunctionalsN. K. Bakirov12 July 2006 | Theory of Probability & Its Applications, Vol. 40, No. 4AbstractPDF (1887 KB)Convergence rates of moments and related functionals in the invariance principle in Banach spacesLithuanian Mathematical Journal, Vol. 31, No. 2 Cross Ref Summary of Reports Presented at Sessions of the Seminar on Probability Theory and Mathematical Statistics at the Institute of Mathematics of the Siberian Section of the USSR Academy of Sciences (September–December 1985)17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 1AbstractPDF (420 KB)An accuracy of Gaussian approximation of sum distribution of independent random variables in Banach spaces19 September 2006 Cross Ref Second-Order Approximation of Random Polygonal Lines in the Donsker-Prokhorov Invariance PrincipleI. S. Borisov and A. A. Borovkov3 August 2006 | Theory of Probability & Its Applications, Vol. 31, No. 2AbstractPDF (1786 KB)The Fiber Method and Its Application to the Study of Functionals of Stochastic ProcessesM. A. Lifshits17 July 2006 | Theory of Probability & Its Applications, Vol. 27, No. 1AbstractPDF (1262 KB)On Conditions of Existence of Bounded Density for Additive Functionals on Some Random ProcessesI. S. Borisov28 July 2006 | Theory of Probability & Its Applications, Vol. 25, No. 3AbstractPDF (818 KB) Volume 21, Issue 2| 1977Theory of Probability & Its Applications History Submitted:27 December 1974Published online:17 July 2006 InformationCopyright © 1977 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1121034Article page range:pp. 283-299ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On the Rate of Convergence for the Invariance Principle

On the Rate of Convergence for the Invariance Principle

A Characterization of the Weak Convergence of Measures

In this paper we shall investigate the so-called weak convergence of measures. Although the origin of the concept of the weak convergence of measures is a probabilistic one, the concept itself is purely measure-theoretical, and should be, therefore, treated by measure-theoretical methods. In Probability Theory the notion of the weak convergence of measures first appeared in Central Limit Problem. Its full importance, however, has been recognized only recently. It is now known as Donsker's Invariance Principle. In this paper we shall follow Prohorov's approach, as presented in [1]. The list of all necessary definitions and results is given in the Introduction. We shall give some conditions for the weak convergence of measures in separable and complete metric spaces, which are expressed in terms of convergence of measures generated in finite dimensional Euclidean spaces. The last convergence can be treated by standard mathematical tools, like the Theory of Fourier Transformations. It should be noted that our theorems concerning the convergence of measures in separable complete metric spaces remain valid if we omit the assumption of completeness. The proofs will remain essentially unchanged; only instead of dealing with compact sets, we should deal with totally bounded closed sets. The theorems given in Section 4 are of interest for the Theory of Stochastic Processes, since they give the conditions for the weak convergence of measures in the functional spaces $D\lbrack 0, 1\rbrack$ and $C\lbrack 0, 1\rbrack$, and to a large class of stochastic processes there correspond measures generated in space $C\lbrack 0, 1\rbrack$, and these measures are usually given in terms of $\mu^{t_1, \cdots, t_m}$, i.e. in terms of finite dimensional distribution functions of the process.