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1
Content available remote A remark on the exact laws of large numbers for ratios of independent random variables
Matuła Przemysław
Probability and Mathematical Statistics
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2022
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Vol. 42, Fasc. 2
219--225
EN
Let (Xn)n∈N and (Yn)n∈N be two sequences of i.i.d. random variable ξ which are independent of each other and all have the distribution of a positive random variable ξ with density fξ . We study weighted strong laws of large numbers for the ratios of the form [wzór]1 in the cases when IEξ = ∞ or limx→0+ fξ (x) = 0 or fξ is unbounded. This research complements some results known so far.
2
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.
3
Content available remote Law of large numbers for monotone convolution
Wang J.-C., Wendler E.
Probability and Mathematical Statistics
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2013
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Vol. 33, Fasc. 2
225--231
EN
Using the martingale convergence theorem, we prove a law of large numbers for monotone convolutions μ1 ◃ μ2 ◃ . . . ◃ μn, where μj ’s are probability laws on R with finite variances but not required to be identical.
4
Content available remote On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables
Pruss A. R.
Bulletin of the Polish Academy of Sciences. Mathematics
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2013
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Vol. 61, no 2
161--168
EN
Let Ω be a countable infinite product Ω1N of copies of the same probability space Ω1, and let {Ξn} be the sequence of the coordinate projection functions from Ω to~Ω1. Let Ψ be a possibly nonmeasurable function from Ω1 to R, and let Xn(ω)=Ψ(Ξn(ω)). Then we can think of {Xn} as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sn=X1+⋯+Xn. By the ordinary Strong Law of Large Numbers, we almost surely have E∗[X1]≤lim infSn/n≤lim supSn/n≤E∗[X1], where E∗ and E∗ are the lower and upper expectations. We ask if anything more precise can be said about the limit points of Sn/n in the nontrivial case where E∗[X1]1], and obtain several negative answers. For instance, the set of points of Ω where Sn/n converges is maximally nonmeasurable: it has inner measure zero and outer measure one.
5
Content available remote Complete convergence under special hypotheses
Stoica G.
Journal of Mathematics and Applications
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2012
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Vol. 35
91-95
EN
We prove Baum-Katz type theorems along subsequences of random variables under Komlós-Saks and Mazur-Orlicz type boundedness hypotheses.
6
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.
7
Content available remote Convergence rates in the law of large numbers for arrays
Kruglov V., Volodina A.
Probability and Mathematical Statistics
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2006
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Vol. 26, Fasc. 1
63--76
EN
In this paper we present new sufficient conditions for complete convergence for sums of arrays of rowwise independent random variables. These conditions appear to be necessary and sufficient in the case of partial sums of independent identically distributed random variables. Many known results on complete convergence can be obtained as corollaries to theorems proved in this paper.
8
Content available remote The completeness in spaces of bounded pettis intergrable functions and in spaces of bounded functions satisfying the law of large numbers
Musiał K.
Demonstratio Mathematica
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2001
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Vol. 34, nr 2
339-344
EN
It has been proven already by Pettis [5] that the space P(, X) of Pettis integrable functions may be non-complete when endowed with the semivariation norm of the integrals. Then Thomas [9] proved that the space is almost always non-complete. In view of the Open Mapping Theorem in such a case no complete equivalent norm can be defined on P(p,,X). The question is now whether there are interesting linear subsets of P(/A, X) where a complete norm does exist. In this paper we consider two such subspaces: the space Poo (/^, X) of scalarly bounded Pettis integrable functions and the space LLNoo(^,X) of scalarly bounded functions satisfying the strong law of large numbers. We prove that in several cases these spaces are complete.
9
Content available remote Cellular Automata and Many-Particle Systems Modeling Aggregation Behavior Among Populations
Morale D.
International Journal of Applied Mathematics and Computer Science
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2000
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Vol. 10, no 1
157-173
EN
A cellular automaton model is presented in order to describe mutual interactions among the individuals of a population due to social decisions.The scheme is used for getting qualitative results, comparable to field experiments carried out on a population of ants which present an aggregative behavior. We also present a second description of a biological spatially structured population of N individuals by a system of stochastic differential equations of Ito type. A 'law of large numbers' to a continuum dynamics described by an integro-differential equation is given.
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