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Questions tagged [characteristic-functions]

Questions about characteristic functions, of a set (which gives $1$ if the element is on the set and $0$ otherwise) or of a random variable (its Fourier transform). Do not use this tag if you are asking about the method of characteristics in PDE or the characteristic polynomial in linear algebra.

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Chf. of a sum of independent r.v. is the product of chfs. Is the converse true?

Let suppose that we have a random variables $X$ and $Y$ whose chfs. can be written as: $$\forall u \in\mathbb{R},\quad\Phi_X(u) = f(u)\Phi_Y(u).$$ Does it mean that we can write $X=Z+Y$ where $Z$ is a ...
NancyBoy's user avatar
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1 answer
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Proving a distribution is not infinitely divisible

I'm trying to show the following: Show that the distribution on $\mathbb R$ with density $f(x) = \frac{1-\cos(x)}{\pi x^2}$ is not infinitely divisible. The characteristic function of this ...
D Ford's user avatar
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3 votes
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29 views

Example of characteristic function which is not regularly varying at zero.

Let $X$ a random variable (in $\mathbb{R}$ for simplicity) and $\varphi(t) = \mathbb{E}[e^{i tX}]$ be its characteristic function. If $X$ has a moment of order $2$, then $|\varphi(t)|^2 = 1 - \frac12 \...
jvc's user avatar
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Estimating simple regression scaling parameter with the characteristic function

For the characteristic function we have: $$\phi_Y(t) = \phi_X(at)$$ where $Y=aX$. Expanding this out we have, \begin{align} \int f_Y(y) e^{ity}dy &= \int f_{aX}(ax)e^{itax} dx = \int f_X(x) e^{...
play's user avatar
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45 views

Reference Request: Probability-and-Convexity Statement

I can prove the following statement, but it would be even better to have a reference I can cite. Does anyone know where this (or a generalization) is proven? Proposition. Let $0 \leq b \leq a$, and ...
CTVK's user avatar
  • 467
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23 views

Inversion theorem implication

So, consider inversion theorem in the form $\forall a, b : P(\xi \in \{a, b\}) = 0 \Rightarrow P(\xi \in [a, b]) = \lim_{T\to \infty} \frac{1}{2\pi} \int_{-T}^T \frac{e^{-iat}-e^{-ibt}}{it} \phi_{\xi}(...
Daniil's user avatar
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Continuity regarding integration over indicator function

i have problems proving the following: Let $D\subset\mathbb{R}^d$ be an open, bounded and simply connected set. Let $f:D\rightarrow\mathbb{R}$ be a continuous and bounded function with $|\nabla f(x)|&...
lleon97's user avatar
1 vote
1 answer
29 views

Solving a stochastic equation by characteristic functions

Based on the work of Nicolas Curien and Takis Konstantopoulos titled Iterating Brownian motions, ad libitum I would like to prove the following (based on the last paragraph of the proof of Proposition ...
user1047209's user avatar
3 votes
1 answer
89 views

Why is the Fourier transform in probability defined with the opposite sign?

For $f \in L^1(\mathbb{R}^n)$ its Fourier transform is defined as $$\hat{f}(\xi) = \int_{\mathbb{R}^n} f(x) e^{-i\xi\cdot x} dx$$ up to a choice of normalization. The inverse Fourier transform is ...
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Find $a_n$ to make $S_n/a_n\Longrightarrow \mathcal N(0,1)$ given $\lim_{t \rightarrow 0} \frac{1-\phi(t)}{t^2 \ln |t|}=-1$.

Let $X_1, X_2, \ldots$ be independent identically distributed random variables with density $f(x)= |x|^{-3}$ if $ |x|>1$, and $f(x)=0$ otherwise, let the characteristic function be $\phi(t):=\...
Ho-Oh's user avatar
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6 votes
1 answer
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Central limit theorem for two-sided Pareto distribution

I am trying to solve the following problem, which provides an example for a central limit theorem in spite of the fact that the variance is infinite. Consider the two-sided Pareto distribution with ...
EnergySkiller's user avatar
2 votes
1 answer
62 views

What's the point of the characteristic function away from zero?

If all I'm interested is in the moments of my random variable $X$, then given its characteristic function $\varphi _{X}(t)$ we have $$ \operatorname {E} \left[X^{n}\right]=i^{-n}\left[{\frac {d^{n}}{...
FriendlyLagrangian's user avatar
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51 views

Proving $\phi(t)=e^{−|t|^{\alpha}}$ is not a characteristic function for α>2.

Problem: Show that the function $\phi(t) = e^{-|t|^{\alpha}}$ is NOT a characteristic function. I have calculated the first two derivatives: $$\phi'(t) = \frac{-\alpha |t|^{\alpha}e^{-|t|^{\alpha}}}{t}...
Melissa's user avatar
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Positivity condition of an integral from two functions

The density operator $\hat{\rho}$ representation by means of symplectic tomogram $\mathcal{W}(X|\mu,\nu)$ (that is a probability density function (pdf) of the quadrature $X$) is the inverse Radon ...
Kim's user avatar
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2X has same distribution as sum of two iid copies of X under smooth transformation

Suppose $f:[0,+\infty)\rightarrow [0,+\infty)$ is a given strictly increasing smooth function starting from $f(0)=0$. I want to figure out under what conditions with respect to $f$ can we find ...
Fangyi Chen's user avatar
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1 answer
57 views

The Laplace transform of a conditional random variable

Let $X$ be exponentially distributed with mean $1$ and $q \in (0,1)$. Define the random variable $Y \triangleq (1-q)X + q$. Now, the CCDF of Y is given by $\mathbb{P}(Y>y) = e^{-\frac{y-q}{1-q}}\...
Mundo's user avatar
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3 votes
1 answer
51 views

Expectation of the indicator function

Define: For $n \geq 0$, on note $X_n=(n+1) \mathbb{1}_{[n+1,+\infty}$, and $\mathcal{F}_n=\sigma(\{1\},\{2\}, \ldots,\{n\},[n+$ $1,+\infty[)$ and $\forall k \in \mathbb{N}^*, \mathbb{P}(\{k\})=\frac{1}...
phi's user avatar
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1 answer
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Conditional characteristic functions and distributions

Let $X$ be an RV, $\mathcal{F}$ a $\sigma$-algebra, $\phi$ the characteristic function of a distribution $\nu$. Assume that for the conditional characteristic function $$\phi_\mathcal{F}(t) := \...
3nondatur's user avatar
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0 votes
1 answer
66 views

How to prove the characteristic function of normal distribution? [closed]

How to prove that the characteristic function of the normal distribution $N(a, σ^2)$ has the form: $$φ (t) = e^{ita - \frac{1}2t^2σ^2}$$ I think that we need to use this formula, but I don't know what ...
Ростислав Романец's user avatar
2 votes
1 answer
17 views

Characteristic functions: upper bound on |phi(t)e^(-itx)|

This post about characteristic functions asks about absolutely integrable $\phi(t)$. This answer mentions "we can find $R$ such that $\int_{\mathbb R\setminus[-R,R]}\lvert\phi(t)\rvert dt<2\pi\...
johnsmith's user avatar
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0 answers
36 views

Derivative of characteristic functional to compute moments

Let $d\mu$ be a Gaussian measure on the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n)$. The characteristic functional of this measure is computed as $$S[f] = \int e^{i\phi(f)}d\mu(\phi), ...
CBBAM's user avatar
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1 answer
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Characteristic function of the mixture of exponential and normal distributions

I've been calculating the characteristic function of a random variable with distribution $\frac{2}{3}E(\frac{1}{3}) + \frac{1}{3}N(6, 3)$. \begin{align} \phi(t) &= \int_{-\infty}^{0} e^{-itx}\frac{...
johnsmith's user avatar
  • 367
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0 answers
41 views

Finding characteristic function of normal distribution for positive numbers only $X$~$N\left(\mu,\sigma ^2\right),E\left[e^{\eta X}\right]=?$

We have: $X$~$N\left(\mu,\sigma^2\right)$ We know that $\eta\ge0$ and thus I need to find $E\left[e^{\eta X}\right]$. So what I thought of doing: $$E\left[e^{\eta X}\right]=E\left[e^{\left(-j\right)j\...
Ben Shaines's user avatar
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1 answer
36 views

Distribution of 2D random vector, a function of the standard normal random vector

The question is to find the distribution of $(\mathbf{a}^TZ, \mathbf{b}^TZ)$ where $Z$ is the standard $n$-variate normal random vector and $\mathbf{a}, \mathbf{b} \in \mathbb{R}^n$ are orthogonal. I'...
johnsmith's user avatar
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1 answer
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Characteristic Function of Multivariate Distribution

Let $f_0, f_1, g_0, g_1\sim\mathcal{N}(0, 1)$ i.i.d. Define $$ \vec h = \begin{pmatrix} h_0\\ h_1 \end{pmatrix} = \begin{pmatrix} f_0g_0 + f_1g_1\\ f_0g_1 +f_1g_0 \end{pmatrix} $$ What is the ...
Mark Schultz-Wu's user avatar
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0 answers
59 views

Decay of the Fourier transform of the indicator of the unit $\ell^p$ ball when $p < 1$.

I am aware of decay rates for the Fourier transform of $B_p(1) = \{x = (x_1, \dots, x_d) \in \mathbb{R}^d|~|x_1|^{p} + \dots + |x_d|^{p} \leq 1\}$, the $\ell^p(\mathbb{R}^d)$ unit ball when $p \geq 1$....
jvc's user avatar
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1 vote
1 answer
65 views

Characteristic functions to evaluate integral

How do you work out $\int e^{itx} \sin^2(t) / t^2 dt$? Given the integral is with respect to $t$, I'm thinking this is about the inversion formula, but the exponent would have to be $-itx$. (In that ...
johnsmith's user avatar
  • 367
0 votes
0 answers
51 views

Probability Density Function of Compound Poisson Process

I am trying to determine if it is possible to compute the probability density function (PDF) of a compound Poisson process $Y(t) = \sum_{i=0}^{N(t)} X_i$, where $N(t)$ is governed by a Poisson process ...
Josh Pilipovsky's user avatar
1 vote
0 answers
42 views

Is there a way to calculate $E\left[\mathrm{sgn}(X)X^n\right]$ if I don't know the pdf for $X$ but I do know the characteristic function?

I need to calculate $E\left[\mathrm{sgn}(X)X^n\right]$, where $\mathrm{sgn}(X)$ denotes the sign of the random variable $X$. I don't know the pdf for $X$, so I can't do the calculation directly. I ...
artag's user avatar
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0 votes
0 answers
53 views

Get PDF from Taylor series of characteristic function

This question represents some thoughts about the following question: How to get PDF from characteristic function In that question $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-itx}\phi(t)dt,$$ ...
eMathHelp's user avatar
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1 vote
0 answers
60 views

Suppose $(X_{n})_{n}$ is series of i.i.d. random variables and $E(X_{1})=2$. Compute $\lim_{n\to\infty} (n \ln (\varphi_{X_{1}}(\frac{2}{n})))$

Let's define $S_{n}=\sum_{k=1}^{n} X_{k}$. According to strong law of large numbers we have $\frac{S_{n}}{n} \rightarrow E(X_{1})=2$ a.s. It follows from that $\frac{S_{n}}{n} \rightarrow 2$ in ...
bnagy01's user avatar
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1 vote
1 answer
99 views

An interesting property of characteristic function

I find the following interesting fact when I study the characteristic functions in probability theory. For any $x\in \mathbb{R}$, the following inequality holds $$ \min_{t\in [1,5]}|1-e^{itx}|\le 1. $$...
user1247096's user avatar
4 votes
1 answer
330 views

Show that two random vectors have the same distribution

Let $W_1,W_2,...$ be independent identically distributed random variables on $[0,\infty).$ Define $T_0=0,T_n=\sum_{k=1}^{n}W_k (n\ge 1).$ Show that, for any $n,m\in\mathbb{Z}^{+}$ and $0\le t_1<\...
Kevin's user avatar
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4 votes
1 answer
165 views

Prove that if characteristic function is absolutely integrable then corresponding density function will be uniformly continuous.

So the point is that you obtain density function through the characteristic function with the inversion formulas. That's the part of the task. Then it supposed to be uniformly continuous. That is, if $...
Egor's user avatar
  • 73
0 votes
0 answers
93 views

Compute standard normal CDF from CDF using Gil-Pelaez inversion formula

I am trying to compute the series representation of the standard normal distribution CDF from its characteristic function. Given a random variable $ X $ following a standard normal distribution with ...
Pierre Cayet's user avatar
1 vote
1 answer
201 views

Gil-Pelaez Formula consistently ends up being equal to 0

For my first post on Math Stackexchange, I ask your help on a specific issue regarding the Gil-Pelaez formula. I have tried various versions of the formula to get the result right but I still cannot ...
Pierre Cayet's user avatar
1 vote
0 answers
62 views

Monotone characteristic function

Let $X$ be a continuous, symmetric random variable such that its characteristic function $\phi_X$ is real, symmetric and with $\lim_{t\to\infty}\phi_X(t)=0$. What other properties must $X$ have in ...
Andrea Aveni's user avatar
4 votes
1 answer
301 views

Characteristic function of product of two random variables with arbitrary normal distributions

I have $X\sim N(0,5)$ and $Y\sim N(1,1)$ components of a gaussian random vector. The covariance of $X$ and $Y$ is 2. I've already proved that $\frac{X}{2}-Y$ is independent from $Y$. I have to ...
Sigma Algebra's user avatar
3 votes
1 answer
240 views

$\phi(t)=\sqrt{1-t^2} \ $ if $|t|<1$ and $\phi(t)=0$ if $|t| \geq 1.$ Prove that $\phi(t)$ is not a characteristic function

$\phi(t)=\sqrt{1-t^2} \ $ if $|t|<1$ and $\phi(t)=0$ if $|t| \geq 1.$ Prove that $\phi(t)$ is not a characteristic function. $\phi(t)$ checks obvious signs of a characteristic function ($\phi(0)=1,...
fragileradius's user avatar
1 vote
1 answer
45 views

Distributions convergence and standard normal random variable using characteristic functions

Suppose $\lambda_{1}$, $\lambda_{2},\ldots\lambda_{n}\ldots$ be a monotone sequence of positive real numbers that go to infinity. Let $X_{n}$ be sequence of random variables with $\Gamma(\lambda_{n},...
maths and chess's user avatar
0 votes
1 answer
44 views

Exchangeability of differentiation and integration to show a characteristic function

I am having a hard time to understand how the exchangeability of integration and differentiation is justified under $E[|X|^{k}]<\infty$ for a characteristic function in the context of showing $E[X^{...
Tucker's user avatar
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1 vote
0 answers
27 views

Evaluate $\mathrm{P}(U \ge 0)$ by knowing the characteristic function of the random variable $U$

Let $X$, $Y$ and $Z$ be independent random variables with characteristic function $$\phi(\theta) = \frac{1}{\sqrt{1+\theta^2}}$$ say whether the r.v. $X+Y+Z$ has a continuous density and evaluate $\...
Ludovico's user avatar
0 votes
1 answer
36 views

Applying Lebesgue’s DCT to prove that finite E|X| implies continuous differentiability

I'd like to ask about bringing the derivative inside the expected value. The context is proving that if $\mathbb{E}X < \infty$, then $\mathbb{E}X = \frac{1}{i}\frac{\mathrm{d}\phi_X(t)}{\mathrm{d}t}...
johnsmith's user avatar
  • 367
0 votes
0 answers
35 views

Uniqueness of probability measures from equality of their moment generating functions in the domain of finiteness of the MGFs.

I encountered a problem in measure theoretic probability which is as follows: Theorem If $\mu_1$ and $\mu_2$ are probability measures on $\mathbb R$ equipped with the Borel $\sigma$-algebra with ...
Kishalay Sarkar's user avatar
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0 answers
48 views

Sufficient condition for the characteristic function of a discrete distribution to be decreasing

Let $X$ be a discrete random variable defined on the lattice such that $P(X \in \mathbb{Z}/h) = 1$, where $\mathbb{Z}/h$ represents the set of integers scaled by a factor of $h$. I wonder if there ...
XiaoHei's user avatar
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0 answers
51 views

characteristic function in definition of a probability density function

I came across this function in a paper (used to compute the distribution of some real-valued random variable $\lambda_{c}$): $$\large f({\lambda})=\sum\limits_{c \in C_{S}} \frac{\chi_{\{\lambda\}}(\...
stucash's user avatar
  • 195
3 votes
1 answer
280 views

Is it true differentiability of characteristic function at 0 implies existence of the first moment.

The context is the following: Prove $|\cos(t)|$ is not a characteristic function of a distribution, while $\cos(t)$ is a characteristic function. I know you can prove it by inverse Fourier ...
Zorualyh's user avatar
  • 802
1 vote
1 answer
49 views

Difference between characteristic functions of two random variables in terms of expection of squared difference

Let $U_1, \ldots, U_n$ be uncorrelated random variables with zero mean and finite variance. Define $X = \sum_{i=1}^n U_i$ and $X_m = \sum_{i=1}^m U_i$ for $m \le n$. Let $\varphi_X(t)$ and $\varphi_m(...
joy's user avatar
  • 1,250
3 votes
1 answer
84 views

Compute the characteristic function of $X\sim\text{Poisson}(Z)$ where $Z$ is exponentially distributed.

The question comes from this post Poisson Process with Randomly Distributed Time. Let $X\sim\text{Poisson}(Z)$ where $Z$ is exponentially distributed. Suppose that $X$ and $Z$ share the same parameter ...
JacobsonRadical's user avatar
1 vote
1 answer
40 views

Inverse the mapping from a subset $B$ of $X$ to its characteristic function $\chi_B: X \to \{0,1\}$

I am trying to find that the the power set P(X) of a set X is isomorphic to the set of maps from X to {0,1} That map should be given by the mapping of a subset $B$ of $X$ to its characteristic ...
darkside's user avatar
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