List questions from characteristic-functions

Published on 07 Nov 2019 - 22:16

70

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Show that $\frac{d}{dt}\Phi_{X_{1}}(0)=a$, then $\Phi_{X_{1}}(t)=1+(ia+o(1))t$ as $t\longrightarrow 0$.

Published on 08 Nov 2019 - 16:58

52

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Show that if $(X_{1}+\cdots+X_{n})/n\rightarrow_{p}a$, then $\Phi_{X_{1}}(t/n)\rightarrow e^{iat}$ as $n\rightarrow\infty$ through integers

Published on 08 Nov 2019 - 19:54

30

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If $X$ has zero mean and finite variance $\sigma^{2}$, show that $|\Phi_{X}(t)-1|\leq 2t^{2}\sigma^{2}$

Published on 10 Nov 2019 - 0:19

770

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Characteristic function of half normal distribution

Published on 15 Nov 2019 - 11:30

834

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Does real characteristic function imply even density

Published on 23 Feb 2026 - 8:19

277

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Showing that $\exp\big(-|x|^p\big)$ is not a characteristic function

Published on 17 Nov 2019 - 8:10

37

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Recover an unknown from sum of i.i.d. RVs.

Published on 20 Nov 2019 - 2:15

182

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Positiveness of some functions, connection with the central limit theorem and stable distributions

Published on 24 Nov 2019 - 6:44

163

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Characteristic function of convolution?

Published on 26 Nov 2019 - 1:49

2.9k

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Deriving the characteristic function of the Cauchy distribution.

Published on 27 Nov 2019 - 0:12

301

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Symmetric difference, characteristic function, Lebesgue norm

Published on 30 Nov 2019 - 19:59

400

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Determine the distribution from characteristic function

Published on 01 Dec 2019 - 23:07

39

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Average concentration function

Published on 02 Dec 2019 - 6:12

578

Views

Characteristic function of sum of squares of Gaussian distributed variables

Published on 02 Dec 2019 - 22:05

List Question

Given characteristic function $\phi(t) = e^{-|t|^{a}/a}$, where a is some constant

Show that $\frac{d}{dt}\Phi_{X_{1}}(0)=a$, then $\Phi_{X_{1}}(t)=1+(ia+o(1))t$ as $t\longrightarrow 0$.

Show that if $(X_{1}+\cdots+X_{n})/n\rightarrow_{p}a$, then $\Phi_{X_{1}}(t/n)\rightarrow e^{iat}$ as $n\rightarrow\infty$ through integers

If $X$ has zero mean and finite variance $\sigma^{2}$, show that $|\Phi_{X}(t)-1|\leq 2t^{2}\sigma^{2}$

Characteristic function of half normal distribution

Does real characteristic function imply even density

Showing that $\exp\big(-|x|^p\big)$ is not a characteristic function

Recover an unknown from sum of i.i.d. RVs.

Positiveness of some functions, connection with the central limit theorem and stable distributions

Characteristic function of convolution?

Deriving the characteristic function of the Cauchy distribution.

Symmetric difference, characteristic function, Lebesgue norm

Determine the distribution from characteristic function

Average concentration function

Characteristic function of sum of squares of Gaussian distributed variables

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