Stability estimates of generalized geometric sums and their applications

Evgueni I. Gordienko

Displaying similar documents to “Stability estimates of generalized geometric sums and their applications”

Comparing the distributions of sums of independent random vectors

Evgueni I. Gordienko (2005)

Kybernetika

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Let $(X_{n}, n \geq 1), ({\tilde{X}}_{n}, n \geq 1)$ be two sequences of i.i.d. random vectors with values in $ℝ^{k}$ and $S_{n} = X_{1} + \dots + X_{n}$ , ${\tilde{S}}_{n} = {\tilde{X}}_{1} + \dots + {\tilde{X}}_{n}$ , $n \geq 1$ . Assuming that $E X_{1} = E {\tilde{X}}_{1}$ , $E | X_{1} |^{2} < \infty$ , $E | {\tilde{X}}_{1} |^{k + 2} < \infty$ and the existence of a density of ${\tilde{X}}_{1}$ satisfying the certain conditions we prove the following inequalities: $v (S_{n}, {\tilde{S}}_{n}) \leq c max {v (X_{1}, {\tilde{X}}_{1}), ζ_{2} (X_{1}, {\tilde{X}}_{1})}, n = 1, 2, \dots,$ where $v$ and $ζ_{2}$ are the total variation and Zolotarev’s metrics, respectively.

On an Estimate of the Remainder in the Central Limit Theorem

Cyril Lenárt (1975)

Matematický časopis

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On the rate of approximation in the random sum CLT for dependent variables

Adhir Kumar Basu (1987)

Aplikace matematiky

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Capital $" O "$ and lower-case $" o "$ approximations of the expected value of a class of smooth functions $(f \in C^{r} (R))$ of the normalized random partial sums of dependent random variables by the expectation of the corresponding functions of Gaussian random variables are established. The same types of approximation are also obtained for dependent random vectors. This generalizes and improves previous results of the author (1980) and Rychlik and Szynal (1979).

Sums of a Random Number of Random Variables and their Approximations with ν- Accompanying Infinitely Divisible Laws

Klebanov, Lev, Rachev, Svetlozar (1996)

Serdica Mathematical Journal

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* Research supported by NATO GRANT CRG 900 798 and by Humboldt Award for U.S. Scientists. In this paper a general theory of a random number of random variables is constructed. A description of all random variables ν admitting an analog of the Gaussian distribution under ν-summation, that is, the summation of a random number ν of random terms, is given. The v-infinitely divisible distributions are described for these ν-summations and finite estimates of the approximation of ν-sum...