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Journal of Theoretical and Applied Mechanics
1999 nr 1 95-108
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Bulletin of the Polish Academy of Sciences. Mathematics
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.
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Probability and Mathematical Statistics
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. T[...]
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Probability and Mathematical Statistics
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 nor[...]
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Probability and Mathematical Statistics
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.
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Probability and Mathematical Statistics
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 we[...]
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Probability and Mathematical Statistics
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 pol[...]
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Probability and Mathematical Statistics
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 som[...]
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Journal of Mathematics and Applications
2009 Vol. 31 51-66
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Probability and Mathematical Statistics
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*[...]
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Probability and Mathematical Statistics
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.
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Probability and Mathematical Statistics
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 artic[...]
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Probability and Mathematical Statistics
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 [...]
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Demonstratio Mathematica
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 s[...]
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Bulletin of the Polish Academy of Sciences. Mathematics
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 B[...]
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Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
2014 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.
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Demonstratio Mathematica
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 per[...]
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