scholarly journals A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 577 ◽  
Author(s):  
Irina Shevtsova ◽  
Mikhail Tselishchev
Keyword(s):  
Type Inequality ◽  
Random Variables ◽  
Independent Random Variables ◽  
Exact Order ◽  
Second Moments ◽  
Generalized Equilibrium ◽  
Kantorovich Distance ◽  
The One ◽  
First Moment ◽  
Geometric Sums

We introduce a generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with finite nonzero first moment and study its properties. In particular, we prove an optimal moment-type inequality for the Kantorovich distance between a distribution and its equilibrium transform. Using the introduced transform and Stein’s method, we investigate the rate of convergence in the Rényi theorem for the distributions of geometric sums of independent random variables with identical nonzero means and finite second moments without any constraints on their supports. We derive an upper bound for the Kantorovich distance between the normalized geometric random sum and the exponential distribution which has exact order of smallness as the expectation of the geometric number of summands tends to infinity. Moreover, we introduce the so-called asymptotically best constant and present its lower bound yielding the one for the Kantorovich distance under consideration. As a concluding remark, we provide an extension of the obtained estimates of the accuracy of the exponential approximation to non-geometric random sums of independent random variables with non-identical nonzero means.

scholarly journals Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables

1999 ◽  
Vol 22 (1) ◽  
pp. 171-177 ◽  
Author(s):  
Dug Hun Hong ◽  
Seok Yoon Hwang
Keyword(s):  
Law Of Large Numbers ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Strong Law ◽  
One Dimensional ◽  
Large Numbers ◽  
Pairwise Independent Random Variables ◽  
The One

Let {Xij}be a double sequence of pairwise independent random variables. If P{|Xmn|≥t}≤P{|X|≥t}for all nonnegative real numbers tandE|X|p(log+|X|)3<∞, for1<p<2, then we prove that∑i=1m∑j=1n(Xij−EXij)(mn)1/p→0    a.s.   as  m∨n→∞.                                     (0.1)Under the weak condition ofE|X|plog+|X|<∞, it converges to 0inL1. And the results can be generalized to anr-dimensional array of random variables under the conditionsE|X|p(log+|X|)r+1<∞,E|X|p(log+|X|)r−1<∞, respectively, thus, extending Choi and Sung's result [1] of the one-dimensional case.

scholarly journals Bernstein-Type Inequality for Widely Dependent Sequence and Its Application to Nonparametric Regression Models

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Aiting Shen
Keyword(s):  
Strong Consistency ◽  
Nearest Neighbor ◽  
Type Inequality ◽  
Random Variables ◽  
Independent Random Variables ◽  
Bernstein Type ◽  
Dependent Random Variables ◽  
Bernstein Type Inequality ◽  
Dependent Sequence ◽  
Fixed Design Regression

We present the Bernstein-type inequality for widely dependent random variables. By using the Bernstein-type inequality and the truncated method, we further study the strong consistency of estimator of fixed design regression model under widely dependent random variables, which generalizes the corresponding one of independent random variables. As an application, the strong consistency for the nearest neighbor estimator is obtained.

The convergence of a sum of independent random variables

1963 ◽  
Vol 59 (2) ◽  
pp. 411-416
Author(s):  
G. De Barra ◽  
N. B. Slater
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Rate Of Convergence ◽  
Radial Distribution Function ◽  
Random Variables ◽  
Covariance Matrices ◽  
Independent Random Variables ◽  
Second Moments ◽  
Image Position ◽  
Real Euclidean Space

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

scholarly journals A POSITIVE TEST FOR FERMI–DIRAC DISTRIBUTIONS OF QUARK–PARTONS

1998 ◽  
Vol 13 (06) ◽  
pp. 441-451 ◽  
Author(s):  
FRANCO BUCCELLA ◽  
OFELIA PISANTI ◽  
LUIGI ROSA ◽  
ILYA DORSNER ◽  
PIETRO SANTORELLI
Keyword(s):  
Large Class ◽  
Pauli Principle ◽  
Positive Test ◽  
The Third ◽  
Second Moments ◽  
The One ◽  
First Moment ◽  
Characteristic Relationship ◽  
Increasing Function ◽  
Inelastic Processes

By describing a large class of deep inelastic processes with standard parametrization for the different parton species, we check the characteristic relationship dictated by Pauli principle: broader shapes for higher first moments. Indeed, the ratios between the second and the first moments and the one between the third and the second moments for the valence partons is an increasing function of the first moment and agrees quantitatively with the values found with Fermi–Dirac distributions.

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (1) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

scholarly journals The uniform strong law of large numbers without any assumption on a family of sets

2020 ◽  
pp. 39-48
Author(s):  
V. Yu. Bogdanskii ◽  
O. I. Klesov
Keyword(s):  
Additional Assumption ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Independent Random Variables ◽  
Strong Law ◽  
Class A ◽  
Large Numbers ◽  
Pairwise Independent Random Variables ◽  
The One ◽  
Mutually Independent

We study the sums of identically distributed random variables whose indices belong to certain sets of a given family A in R^d, d >= 1. We prove that sums over scaling sets S(kA) possess a kind of the uniform in A strong law of large numbers without any assumption on the class A in the case of pairwise independent random variables with finite mean. The well known theorem due to R. Bass and R. Pyke is a counterpart of our result proved under a certain extra metric assumption on the boundaries of the sets of A and with an additional assumption that the underlying random variables are mutually independent. These assumptions allow to obtain a slightly better result than in our case. As shown in the paper, the approach proposed here is optimal for a wide class of other normalization sequences satisfying the Martikainen–Petrov condition and other families A. In a number of examples we discuss the necessity of the Bass–Pyke conditions. We also provide a relationship between the uniform strong law of large numbers and the one for subsequences.

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (01) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

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