Sufficient conditions for the weak convergence of the distributions of the random variables (1 - x)xi(x) as x -> 1- to the limiting gamma-distribution are put forward. The random variable e x has power-series distr...
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Sufficient conditions for the weak convergence of the distributions of the random variables (1 - x)xi(x) as x -> 1- to the limiting gamma-distribution are put forward. The random variable e x has power-series distribution with radius of convergence 1 and parameter x is an element of (0, 1). Limit theorems for the probabilities P{xi(x )= k} are proposed. Asymptotic expansions of local probabilities are derived for sums of independent identically distributed variables with the same distribution as xi(x) in a triangular array with x -> 1-. For the corresponding general allocation scheme, local limit theorems for the joint distributions of the occupancies of the cells are obtained.
A strong law is proved for weighted sums S-n = Sigma a(in)X(i) where X(i) are i.i.d. and {a(in)} is an array of constants. When sup(n(-1) Sigma\a(in)\(q))(1/q) (a.s) 0. When q = infinity this reduces to a result of Ch...
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A strong law is proved for weighted sums S-n = Sigma a(in)X(i) where X(i) are i.i.d. and {a(in)} is an array of constants. When sup(n(-1) Sigma\a(in)\(q))(1/q) < infinity, 1 < q less than or equal to infinity and X(i) are mean zero, we show E\X\(p) < infinity, p(-1) + q(-1) = 1 implies S-n/n -->(a.s) 0. When q = infinity this reduces to a result of Choi and Sung((1)) who showed that when the {a(in)} are uniformly bounded, EX = 0 and E\X\ < infinity implies S-n/n -->a.s. 0. The result is also true when q = 1 under the additional assumption that lim sup \a(i)n\ n(-1) log n = 0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a(in)} are uniformly bounded, E\X\(1/alpha) < infinity implies S-n/n(alpha) -->(a.s.) 0 for alpha > 1, but this is not true in general for 1/2 < alpha < 1, even when the X(i) are symmetric. In that case the additional assumption that (x(1/alpha)log(1/alpha-1 x) P(\X\ greater than or equal to x) --> 0 as x up arrow infinity provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a(in)}.
Abstract: In a recent book, Parthasarathy provides limit theorems for sums of independent random variables defined on a metrizable locally compact abelian group. These results make heavy use of the metric assu...
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Abstract: In a recent book, Parthasarathy provides limit theorems for sums of independent random variables defined on a metrizable locally compact abelian group. These results make heavy use of the metric assumption. This paper consists of a reworking of certain results contained in Parthasarathy to see what can be done without the metric restriction. Among the topics considered are: necessary and sufficient conditions for a limit law to have an idempotent factor; the relationship between limits of compound Poisson laws and limits of sums of independent random variables; and a representation theorem for certain limit laws.
We consider the Cox-Ross-Rubinstein model of option prices which is a simple binomial model and deal with its multivariate extensions. The model consists of n independent up or down movements of the (multivariate) pri...
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We consider the Cox-Ross-Rubinstein model of option prices which is a simple binomial model and deal with its multivariate extensions. The model consists of n independent up or down movements of the (multivariate) price. We discuss the model in the view of the limiting distributions for the price as well for the extreme changes of the prices during a period T which is split up into n small price changes, which depend on n (with nh = T). Interesting is also whether the components of the prices and of the extremes are asymptotically dependent.
On any aperiodic measure preserving system, there exists a square integrable function such that the associated stationary process satifies the Almost Sure Central Limit Theorem.
On any aperiodic measure preserving system, there exists a square integrable function such that the associated stationary process satifies the Almost Sure Central Limit Theorem.
In this paper, we present more analysis about the finiteness of the first meeting time between Gaussian jump and Brownian particles in the fluid. Rather than using the uniform integrability conditions which are detail...
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In this paper, we present more analysis about the finiteness of the first meeting time between Gaussian jump and Brownian particles in the fluid. Rather than using the uniform integrability conditions which are detailed in El-Hadidy [Mod. Phys. Lett. B 33(22) (2019) 1950256], we show the triangular arrays of the random sequence that represents the probable positions of the first meeting at any time t is an element of R+, to get the stronger moment conditions which are sufficient for the finiteness of this time.
Let {V-i, (j);(i, j) is an element of N-2} be a two-dimensional array of independent and identically distributed random variables. The limit laws of the sum of independent random products Zn = (Nn)Sigma(i=1) (n)Pi(j=1...
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Let {V-i, (j);(i, j) is an element of N-2} be a two-dimensional array of independent and identically distributed random variables. The limit laws of the sum of independent random products Zn = (Nn)Sigma(i=1) (n)Pi(j=1) e(Vi,j) as n, N-n -> infinity have been investigated by a number of authors. Depending on the growth rate of N-n, the random variable Z(n) obeys a central limit theorem or has limiting alpha-stable distribution. The latter result is true for non-lattice V-i,V-j only. Our aim is to study the lattice case. We prove that although the (suitably normalized) sequence Z(n) fails to converge in distribution, it is relatively compact in the weak topology, and we describe its cluster set. This set is a topological circle consisting of semi-stable distributions.
Abstract: Some results are obtained concerning the convergence in distribution of the row sums of a triangular array of certain dependent random variables. The form of dependence considered is that of martinga...
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Abstract: Some results are obtained concerning the convergence in distribution of the row sums of a triangular array of certain dependent random variables. The form of dependence considered is that of martingales within rows, and the results are obtained under conditions which parallel those of the classical case of convergence in distribution, to infinitely divisible laws with bounded variances, of the row sums of elementary systems of independent random variables.
In this paper it will be shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable rando...
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In this paper it will be shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable random variables.
Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient condi...
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Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.
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