A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums
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Tran Loc Hung
Abstract
The purpose of this paper is to study a chi-square-type distribution who degrees of freedom are geometric random variables in connection with weak limiting distributions of geometric random sums of squares of independent, standard normal distributed random variables. Some characteristics of chi-square-type random variables with geometrically distributed degrees of freedom including probability density function, probability distribution function, mean and variance are calculated. Some asymptotic behaviors of chi-square-type random variables with geometrically distributed degrees of freedom are also established via weak limit theorems for normalized geometric random sums of squares of independent, standard normal distributed random variables. The rates of convergence in desired weak limit theorems also estimated through Trotter’s distance. The received results are extensions and generalizations of several known results.
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Communicated by Gejza Wimmer
6 Appendix
We must recall that Proposition 2.1 in Section 2 could be directly proved as follows:
Let X be a random variable with 𝔼(|X|k) < +∞. Then 𝔼(|X|j) < +∞, for any 1 ≤ j ≤ k, and
Proof
We shall begin with showing that
Consider the first term of right-side of (6.1), since |X| < 1 for j ≥ 1 and 𝕀(|X| ≤ 1) ≤ 1, it follows that
Hence
According to second term of right-side of (6.1), since |X| > 1 and 1 ≤ j ≤ k, we have
Thus, for 1 ≤ j ≤ k,
Moreover, it follows that
Finally, from (6.1), for each random variable X, if 1 ≤ j ≤ k, it may be concluded that
From above inequality, if 𝔼|X|k < +∞, then 𝔼|X|j < +∞, for 1 ≤ j ≤ k. The proof is complete. □
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© 2020 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Doc. RNDr. Roman Frič, DrSc. – 3/4 C?
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- Two by two squares in set partitions
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- On the topology of partial metric spaces
- The trace of the commutator of the Cesàro operator and a compact operator
- Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold
- On 4-th root metrics of isotropic scalar curvature
- On ultrametric-preserving functions
- Porosity of 𝓢-approximately continuous functions
- On a modified Burr XII distribution having flexible hazard rate shapes
- A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums
- An alternative estimate for the numerical radius of Hilbert space operators
Artikel in diesem Heft
- Regular papers
- Doc. RNDr. Roman Frič, DrSc. – 3/4 C?
- Sugihara algebras and Sugihara monoids: Multisorted dualities
- Two by two squares in set partitions
- Identifying quantum logics by numerical events
- Some remarks on divisible polyhedral MV-algebras
- Remarks on b-metrics, ultrametrics, and metric-preserving functions
- Estimates of essential norm of generalized weighted composition operators from Bloch type spaces to nth weighted type spaces
- Maximum term of transcendental entire function and spider’s web
- Entire solutions of certain type of non-linear differential equations
- On the oscillatory behavior of third order differential equations with a sublinear neutral term
- Eigenvalue problems for a class of nonlinear Hadamard fractional differential equations with p-Laplacian operator
-
The amalgam space
- On the topology of partial metric spaces
- The trace of the commutator of the Cesàro operator and a compact operator
- Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold
- On 4-th root metrics of isotropic scalar curvature
- On ultrametric-preserving functions
- Porosity of 𝓢-approximately continuous functions
- On a modified Burr XII distribution having flexible hazard rate shapes
- A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums
- An alternative estimate for the numerical radius of Hilbert space operators
