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1
Content available remote Limit theorems for a higher order time dependent Markov chain model
Kokoszka Piotr, Kutta Tim, Singh Deepak, Wang Haonan
Probability and Mathematical Statistics
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2023
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Vol. 43, Fasc. 1
121--139
EN
The paper establishes a strong law of large numbers and a central limit theorem for a sequence of dependent Bernoulli random variables modeled as a higher order Markov chain. The model under consideration is motivated by problems in quality control where acceptability of an item depends on the past k acceptability scores. Moreover, the model introduces dependence that may evolve over time and thus advances the theory for models with time invariant dependence. We establish explicit assumptions that incorporate this dynamic dependence and show how it enters into the limits describing long-term behavior of the system.
2
Content available remote bm-Central Limit Theorems associated with non-symmetric positive cones
Oussi Lahcen, Wysoczański Janusz
Probability and Mathematical Statistics
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2019
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Vol. 39, Fasc. 1
183--197
EN
Analogues of the classical Central Limit Theorem are proved in the non-commutative setting of random variables which are bm-independent and indexed by elements of positive non-symmetric cones, such as the circular cone, sectors in Euclidean spaces and the Vinberg cone. The geometry of the cones is shown to play a crucial role and the related volume characteristics of the cones is shown.
3
Content available remote On the number of reflexive and shared nearest neighbor pairs in one-dimensional uniform data
Bahadir S., Ceyhan E.
Probability and Mathematical Statistics
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2018
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Vol. 38, Fasc. 1
123--137
EN
For a random sample of points in R, we consider the number, of pairs whose members are nearest neighbors (NNs) to each other and the, number of pairs sharing a common NN. The pairs of the first type are called, reflexive NNs, whereas the pairs of the latter type are called shared NNs. In, this article, we consider the case where the random sample of size n is from, the uniform distribution on an interval. We denote the number of reflexive NN pairs and the number of shared NN pairs in the sample by Rn and Qn, respectively. We derive the exact forms of the expected value and the variance for both Rn and Qn, and derive a recurrence relation for Rn which may also be used to compute the exact probability mass function (pmf) of Rn. Our approach is a novel method for finding the pmf of Rn and agrees with the results in the literature. We also present SLLN and CLT results for both Rn and Qn as n goes to infinity.
4
Content available remote Cramér type large deviations for trimmed L-statistics
Gribkova N.
Probability and Mathematical Statistics
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2017
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Vol. 37, Fasc. 1
101--118
EN
In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those in Callaert et al. (1982) – the first and, as far as we know, the single article where some results on probabilities of large deviations for the trimmed L-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed L-statistic by a non-trimmed L-statistic (with smooth weight function) based onWinsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed L-statistics under quite mild and natural conditions.
5
Content available Central Limit Theorem visualized in Excel
Hodaňová J., Zdráhal T.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2014
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Vol. 19
51--55
EN
The Central Limit Theorem states that regardless of the underlying distribution, the distribution of the sample means approaches normality as the sample size increases. This paper describes the steps in MS Excel to help students’ better understanding of this theorem.
6
Content available remote The limiting behaviour of sums and maximums of iid random variables from the viewpoint of different observers
Krajka T. K., Rychlik Z.
Probability and Mathematical Statistics
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2014
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Vol. 34, Fasc. 2
237--252
EN
The limiting behaviour of observed and all random variables in the max limit schema was considered by Mladenović and Piterbarg (2006) and Krajka (2011). Here those results are generalised in two directions: we allow more than one observer and one superobserver; we consider the max limit schema as well as the sum limit schema.
7
Content available remote A Wick functional limit theorem
Parczewski P.
Probability and Mathematical Statistics
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2014
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Vol. 34, Fasc. 1
127--145
EN
We prove that weak convergence of multivariate discrete Wiener integrals towards the continuous counterparts carries over to the application of discrete and continuous Wick calculus. This is done by the representation of arbitrary Wick products of Wiener integrals in terms of generalized Hermite polynomials and a discrete analog of the Hermite recursion. The result is a multivariate non-central limit theorem in the form of a Wick functional limit theorem. As an application we give approximations of multivariate processes based on fractional Brownian motions for arbitrary Hurst parameters H ∈ (0, 1).
8
Content available remote Principle of Conditioning revisited
Jakubowski A.
Demonstratio Mathematica
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2012
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Vol. 45, nr 2
325-336
EN
Principle of Conditioning is a well known heuristic rule which allows constructing limit theorems for sums of dependent random variables from existing limit theorems for independent summands. In the paper we state a general limit theorem on converegnce to stable laws, which is valid for stationary sequences and provides a link between the Principle of Conditioning and ergodic theorems.
9
Content available remote Limit theory for planar Gilbert tessellations
Schreiber T., Soja N.
Probability and Mathematical Statistics
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2011
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Vol. 31, Fasc. 1
149--160
EN
A Gilbert tessellation arises by letting linear segments (cracks) in R2 unfold in time with constant speed, starting from a homogeneous Poisson point process of germs in randomly chosen directions. Whenever a growing edge hits an already existing one, it stops growing in this direction. The resulting process tessellates the plane. The purpose of the present paper is to establish a law of large numbers, variance asymptotics and a central limit theorem for geometric functionals of such tessellations. The main tool applied is the stabilization theory for geometric functionals.
10
Content available remote Limit theorems for products of sums of independent random variables
Krajka T. K., Rychlik Z.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 1
73--85
EN
Let {Xn; n ≥ 1} be a sequence of independent random variables with finite second moments and {Nn; n ≥ 1} be a sequence of positive integer-valued random variables. Write Sn. =Σnk-1(Xk−EXk); n ≥1; and let N be a standard normal random variable. In the paper the convergences...[formula], are considered for some sequences {an} and {γn} of positive integer numbers such that Sn + an≥ 0 a.e. The case when γn are random variables is also considered. The main results generalize the main theorems presented by Pang et al. [3].
11
Content available remote A central limit theorem for multivariate strongly mixing random fields
Tonelli C.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 2
215--222
EN
In this paper we extend a theorem of Bradley under interlaced mixing and strong mixing conditions. More precisely, we study the asymptotic normality of the normalized partial sum of an α-mixing strictly stationary random field of random vectors, in the presence of another dependence assumption.
12
Content available remote Optimality of the auxiliary particle filter
Douc R., Moulines É, Olsson J.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 1
1--28
EN
In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) proposed by Pitt and Shephard [17]. Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. By examining the recursive formula for the asymptotic variance of the CLT we identify first-stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved.
13
Content available remote The speed of convergence of random products of sums of independent random variables
Krajka T., Rychlik Z.
Journal of Mathematics and Applications
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2009
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Vol. 31
51-66
14
Content available remote Asymptotics for products of a random number of partial sums
Przystalski M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2009
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Vol. 57, no 2
163-167
EN
We consider products of a random number of partial sums of independent, identically distributed, positive square-integrable random variables. We show that the distribution of these products is asymptotically lognormal.
15
Content available remote Some remarks on the central limit theorem for functionals of linear processes under short-range dependence
Furmańczyk K.
Probability and Mathematical Statistics
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2007
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Vol. 27, Fasc. 2
235--245
EN
In this paper we consider the central limit theorems for functionals G: Rm-> R of one-sided m-dimensional linear processes Xt=∑∞r=0 where Ar is a nonrandom matrix mxm and Zt’s are i.i.d. random vectors in Rm.
16
Content available remote Finite difference equations and convergence rates in the central limit theorem
Lindhagen L.
Probability and Mathematical Statistics
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2007
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Vol. 27, Fasc. 2
153--166
EN
We apply the theory of finite difference equations to the central limit theorem, using interpolation of Banach spaces and Fourier multipliers. Let S*n be a normalized sum of i.i.d. random vectors, converging weakly to a standard normal vector N. When does ǁEg (x + S*n) -E g (X + N)ǁLp(dx)tend to zero at a specified rate? We show that, under moment conditions, membership of g in various Besov spaces is often sufficient and sometimes necessary. The results extend to signed probability.
17
Content available remote Convergence rate in CLT for vector-valued random fields with self-normalization
Bulinski A., Kryzhanovskaya N.
Probability and Mathematical Statistics
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2006
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Vol. 26, Fasc. 2
261--281
EN
Statistical version of the central limit theorem (CLT) with random matrix normalization is established for random fields with values in a space Rk (k ≥ 1). Dependence structure of the field under consideration is described in terms of the covariance inequalities for the class of bounded Lipschitz ”test functions” defined on finite disjoint collections of random vectors constituting the field. The main result provides an estimate of the convergence rate, over a family of convex bounded sets, in the CLT with random normalization.
18
Content available remote On the almost sure central limit theorems for the vectors of several large maxima and for some random permanents
Dudziński M.
Demonstratio Mathematica
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2006
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Vol. 39, nr 4
949-963
EN
In our paper we prove two kinds of the so-called almost sure central limit theorem (ASCLT). The first one is the ASCLT for the vectors ((Mn(1) , . . . ,Mn(r)), where Mn(j) n - the j-th largest maximum of X1, . . . ,Xn and {Xi} is an i.i.d. sequence. Our second result is the ASCLT for some random permanents.
19
Content available remote On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers
Przytycki F.
Bulletin of the Polish Academy of Sciences. Mathematics
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2006
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Vol. 54, no 1
41-52
EN
We prove that the hyperbolic Hausdorff dimension of Fr Ω, the boundary of the simply connected immediate basin of attraction Ω to an attracting periodic point of a rational mapping of the Riemann sphere, which is not a finite Blaschke product in some holomorphic coordinates, or a 2 : 1 factor of a Blaschke product, is larger than 1. We prove a "local version" of this theorem, for a boundary repelling to the side of the domain. The results extend an analogous fact for polynomials proved by A. Zdunik and relies on the theory elaborated by M. Urbanski, A. Zdunik and the author in the late 80-ties. To prove that the dimension is larger than 1, we use expanding repellers in δΩ constructed in [P2]. To reach our results, we deal with a quasi-repeller, i.e. the limit set for a geometric coding tree, and prove that the hyperbolic Hausdorff dimension of the limit set is larger than the Hausdorff dimension of the projection via the tree of any Gibbs measure for a Holder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Holder potentials on Julia sets.
20
Content available remote Data-driven score test of fit for conditional distribution in the GARCH (1, 1) model
Inglot T., Stawiarski B.
Probability and Mathematical Statistics
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2005
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Vol. 25, Fasc. 2
331--362
EN
A data-driven score test for a conditional distribution in the GARCH (1, 1) model is proposed. Conditional distribution assumption is verified by a score test, obtained from nesting the null density into an exponential family and then choosing the dimension of this exponential family by a score-based selection rule. A simulation study, which is provided, shows good empirical behaviour of the proposed test, outperforming in most cases the behaviour of competitive tests.
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