Questions tagged [characteristic-function]

21 questions with no upvoted or accepted answers
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Characteristic function inequality

Random variable $X$ and its characteristic function $\phi_X(t)$ then $$\Pr\left(|X|>\frac2T\right) \leq 2\left(1 - \frac1{2T}\int_{-T}^{T}\phi_X(t)dt\right) $$ I cannot find a way how to ...
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154 views

Sum of truncated Gammas and degenerate

I have a variable $X$ which I am modelling with a mixture model: $$\begin{aligned} (X|A) &\sim \mathbb{1}_{0 \leq x < w \cdot m} \cdot \frac{\text{Gamma}(\alpha,0,\beta / m)}{k_1} \\ (X|B) &...
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478 views

What is the distribution of an affine transformation of a log-normal variable?

Let $X$ be a log-normal variate and $Y = aX + b$ is the affine transformation X. Is $Y$ log-normal? I suspect it is not. Since $X$ is log-normal, its expected value is $$ E[X] = \exp(M + S^2/2) $$ ...
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246 views

Characteristic functions can establish stochastic dominance?

In an answer to the question here What is the purpose of characteristic functions? people answered the general question about characteristic functions. One answer mentioned that one can use it to ...
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23 views

finding process corresponding to laplace transform

I have a positive stochastic process $X(t)$ with Laplace transforms $$ \mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b} $$ One can clearly see that the ...
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37 views

Good book on characteristic functions that includes the CF-proof of the CLT

The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ...
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104 views

How can you determine if a function is heavy tailed from its characteristic function?

The question is given as follow: Let $N$ have a Poisson distribution with mean $\lambda$. $X_i$ is Cauchy distribution with mode 0 and and scaling parameter $1$. Find the characteristic function ...
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34 views

Why do we use exponent in characteristic function?

A student who is attending probability 101, learned about normal distribution and generating functions recently. We are given a "generating function" as follows: $$G(t)=<e^{itx}>=\int_{-\infty}^...
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24 views

How can we write the below characteristic function?

Let us assume that $X$ is a random variable and $a$ is a constant. Now suppose $Y=a+bX$, what would the characteristic function of $Y$ would be? Is it? \begin{eqnarray} \mathbb{E}_X\left[\exp(iuY)\...
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27 views

On likelihood functions and characteristic functions

Let me preface this by saying that if someone manages to provide a solution to my problem, I will forever be indebted to them, as this problem has driven me crazy. Let us first assume that the ...
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71 views

A simple question about expectations

@psboonstra This is a valid point. After posting it, I too realized that the question is ill-posed. I attempted to oversimplify a problem that I had encountered in finding a characteristic function of ...
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79 views

Question about the log-normal distribution

The main object of my question is this: if $X$ has a log-normal distribution, $Y = X + Z$ and $Y$ has the same distribution as that of $Z^2$ (in other words, $F_{Z^2} = F_{X+Z}$) and $X, Z$ are ...
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39 views

Difference between characteristic function and F-transform

I'm struggling to understand the difference between this two functions. I have this condition: $P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\...
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171 views

How to pass from {Probability density function, convolution} to {Probability density function, characteristic function}?

In Forsman, W.C. (1986) "Polymers in solution: theoretical considerations and newer methods of characterization", Springer, New York. https://www.springer.com/la/book/9780306421464 page 24, it states:...
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37 views

Sufficient statistic from characteristic function?

I have a density function $f(;\theta):{\mathbb R}\rightarrow{\mathbb R}_+$, where $\theta\in{\mathbb R}^d$. I know that there is a sufficient statistic $T$ of dimension $d$. If $\varphi_n$ is the ...
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26 views

Joint cumulants of Zn2 characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \...
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651 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
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19 views

Goodness of fit using characteristic function

When assessing goodness of fit of a model, one can use, for example, a Q-Q plot of empirical vs theoretical distributions. But how does one perform a GoF assessment when there is no closed form ...
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20 views

How to find the characteristic function of a function related with Shannon entropy?

A random variable X is distributed with a known probability distribution $p(x)$. Suppose that $x$ is sampled in an independent and identically distributed process and with the results $\vec{x}=(x_1,...
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7 views

Question on excluding one point on the support of a continuous random variable

Suppose that I have a continuous random variable $X$ which has a support equal to $\mathbb{R}$. Can I construct a new random variable $Y$ such that it is equal in distribution to $X$ on the support of ...
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35 views

Classification of random variables with (non-)vanishing characteristic function

What is a necessary and sufficient condition for a random variable to have non-vanishing characteristic function? I wonder how should one check whether a random variable's characteristic function is ...