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.

Filter by
Sorted by
Tagged with
0
votes
0answers
29 views

Square integrability of charateristic function

For a measure $\mu$ on $\mathbb{R}$, the characteristic function $\varphi_\mu: \mathbb{R}\to[0,1]$ is defined as $$\varphi_\mu(t)=\mathbb{E}_{X\sim\mu}[e^{i tX }] $$ I understand that $\int_\mathbb{R}|...
2
votes
0answers
34 views

Poisson sum is a mixture model?

I know that the sum of two independent normal random variables is normal. Particulary, when one is copy of other, i.e., if $X_1, X_2 \sim \mathcal{N}(0,\sigma^2)$, independent, we have: $$X_1 + X_2 \...
0
votes
0answers
23 views

Characteristic function of a random variable: what is the argument/input?

The characteristic function (c.f.) of a random variable, $X$, is defined as: $$ \varphi_X(t) = \mathbb{E}[e^{i\langle t,x \rangle}] $$ For example, the c.f. of a Poisson random variable, $X$ is: $$ \...
2
votes
2answers
59 views

Show $\mu(A)= \alpha \int_A x^2 dF(x)$ for a compound Poisson process

The question is suppose that $N,Y_1,Y_2,....,Y_n$ are independent and $Y_n$ has the common distribution function $F$ and $N$ has the Poisson distribution with mean $\alpha$. If $F$ has mean $0$ and ...
3
votes
1answer
54 views

Mean and characteristic function of $Y=\sum_{n=1}^{\infty}\prod_{k=1}^{n}X_k$

I have problems with this exercise. Let $X_1, X_2, \ldots $ r.v. independent and equally distributed exponential with parameter $\lambda > 1$. Verify if random variable $$Y=\sum_{n=1}^{\infty}\...
1
vote
1answer
26 views

Find the characteristic polynomial of the matrix $M$ in terms of characteristic polynomial of $N$.

Find the characteristic polynomial of the matrix $M$ in terms of characteristic polynomial of $N$. $M=\begin{pmatrix}N+V & U\\U^T& I\end{pmatrix}$ where $V=\left[ \begin{array}{ccccc} n &...
0
votes
1answer
45 views

tinuity theorem

Let $X_n$ be $Po(n)$-distributevy's continuity theorem but I don't really know how.
0
votes
0answers
31 views

Limit property of characteristic function [closed]

I have some problems solving this exercise and hope someone can give me a hint. Let $\varphi$ be the characteristic function of an absolute-continuous random variable $X$. Show that: \begin{align*} \...
0
votes
2answers
43 views

Show that : $1 - |\phi(t)| \ge \frac{1-|\phi(2t)|}{4}$ [closed]

Show that : $1 - |\phi(t)| \ge \frac{1-|\phi(2t)|}{4}$ where $\phi(t)$ is a characteristic function . I am able to prove another part of the question $\Re(1-\phi(t)) \ge \Re(\frac{1-\phi(2t)}{4})$ . ...
1
vote
1answer
59 views

Weak convergence of a sequence $(f_n) \subset L^2(\mathbb{R})$ and convergence of $\chi_n f_n$

I am asked to solve the following problem. Let $(f_n)_n \subset L^2(\mathbb{R})$ be such that $f_n \rightharpoonup f$ in $L^2(\mathbb{R})$. Let $I_n = (-n, n)$ and denote with $\chi_n$ the ...
2
votes
1answer
91 views

How can I get distribution function from characteristic function?

Suppose $F=F(x)$ is distribution function of r.v. $X$ and its characteristic function is $\varphi_X(t)=\int_{-\infty}^\infty e^{itx}dF(x)$. Then for any $a<b$ where $F$ is continuous at $a,b$ we ...
0
votes
0answers
25 views

Approximation of characteristic function

I have bene asked to show that given a bounded interval [a,b] the characteristic function of said interval can be approximated in norm $L^p$ ($1≤p<\infty$) by differentiable functions with compact ...
4
votes
1answer
64 views

When can limit of a sequence of characteristic be also a characteristic function?

Say {$\phi_n$} be a sequence of characteristic function with some densities {$f_n$} that converges pointwise to $\phi$ , i.e., $\lim_{n\to\infty} \phi_n (t) = \phi(t) \forall t\in \mathbb{R}$ . If $\...
1
vote
0answers
32 views

Stable laws of probabilty and convergence.

I'm working on a probabilty exercise and i'm stuck at some point. I did the first 3 questions without trouble. Here is what is says : We consider $(\Omega,\mathcal{A},\mathbb{P}) $ a probability ...
2
votes
1answer
50 views

Bernstein's Theorem (Probability)

I am doing an exercise that establishes the Bernstein's Theorem (it's called that way in french i don't know if it is known with this name in english). It states the following : Let $(\Omega,\mathcal{...
0
votes
0answers
20 views

Find E[e^(aX^4)], where X ~ N(0, sigma^2)

I have to compute $\mathbb{E}\{e^\left(aX^4\right)\}$, where $X \sim \mathcal{N}(0, \sigma^2)$ is a normal random variable. Following the top answer here, I tried to find the solution. This requires ...
0
votes
0answers
15 views

even continuous function that is convex and vanishes at infinity and has value $1$ at $0$ is a characterisitc function

I would like to get some feedback on the answer I wrote below: The problem is from a textbook I'm reading: problem 3.2.8 of Probability Theory Lecture Notes by Panchenko. I posted about 3.2.7 here ...
1
vote
0answers
42 views

Find characteristic function of the law with density $\max(1-|x|,0)$

I would like to get some feedback and hints for my answer below: The problem is from a textbook I'm reading: problem 3.2.7 of Probability Theory Lecture Notes by Panchenko. Here's the question: Find ...
2
votes
1answer
37 views

Showing an identity of two uniform distributed random variables by using characteristic functions and the inversion formula

I have to show that for $a,b > 0$ $$\int_\mathbb{R} \dfrac{\sin(at)\sin(bt)}{t^2}dt = \pi\min(a,b)$$ by using characteristic functions and the inversion formula. We do have the hint that we should ...
0
votes
1answer
27 views

Inversion formula and characteristic functions for a point mass

Durett Probability Theory and Examples suggest that the following inversion result (p.95) is intuitive. However, I cannot figure out how to prove it. Here is the result : If $X$ has characteristic ...
0
votes
2answers
83 views

A difficult Integral Question

How to calculate $$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{t^2}{2x^2}-\frac{x^2}{2}\right)\,dx$$ Hi, this is a result of a characteristic function problem calculating the ...
3
votes
0answers
49 views

Deriving the characteristic function of the Pareto distribution

For the Pareto probability distribution, with density function $$f(x ; x_m, \alpha) = \begin{cases} 0 & \qquad \text{for $x \leq 0$} \\ \alpha \frac{x_m^\alpha}{x^{\alpha+1}} & \qquad \text{...
5
votes
2answers
109 views

Weighted summation of symmetric Bernoulli RV. Characteristic function inequality

Let $$S_n = \sum_{k=1}^n \frac{X_k}{\sqrt{k}}$$ where $X_1, X_2, \ldots$ are iid symmetric Bernoullis with parameter $\frac{1}{2}$: $$X_k = \begin{cases} 1 &p=\frac{1}{2}\\ -1 &p=\frac{1}{2} \...
1
vote
1answer
46 views

Convergence of characteristic function

Let $$S_n = \sum_{k=1}^n \frac{X_k}{\sqrt{k}}$$ where $X_1, X_2, \ldots$ are iid symmetric Bernoullis with parameter $\frac{1}{2}$: $$X_k = \begin{cases} 1 &p=\frac{1}{2}\\ -1 &p=\frac{1}{2} \...
0
votes
0answers
18 views

The weak limit of a sequence of infinitely divisible probability measures is infinitely divisible

Let $\{\mu_n\}_n$ be a sequence of infinitely divisible probability measures on $\mathbb{R}^d$ with $\mu_n \xrightarrow{\text{weak}} \mu$ for some probability measure $\mu$. I want to prove that $\mu$ ...
1
vote
0answers
22 views

$g = \chi_{E}$ where $E = \{(x, t): t \ge f(x)\}$, a function in two variables, is measurable

Equip $[0, \infty)$ with the Borel $\sigma$-algebra. Let $(X, \mathcal{A}, \mu)$ be a $\sigma$-finite measure space. Let $f: X \to [0, \infty)$ be a measurable function. Define $g: X \times [0, \infty)...
0
votes
1answer
22 views

Integrating some characteristic functions

How would I go about computing the following function: $$F(x_1,x_2) = \int_0^1 \chi_{[0,1]^2}(x_1 - t, x_2-t) \ dt$$ My idea is to observe that $$F(x_1, x_2) = \int_0^1 \chi_{[0,1]} (x_1-t) \chi_{[0,1]...
2
votes
0answers
35 views

Characteristic function of non-Gaussian Ornstein-Uhlenbeck process

I want to calculate the characteristic function $\phi(u)=\mathbb{E}\left[\exp(iuX_t)\right]$ of the following superposition of non-Gaussian Ornstein-Uhlenbeck process $$X_t=\int_0^t\left[e^{-a(t-s)}-e^...
2
votes
0answers
38 views

How to find or approximate probability distribution from known values of the characteristic function?

If I have a known discrete values for the characteristic function (I know $a_n$ values for specific values of $\omega_n = \frac{2\pi i n}{d}$): $$a_n = \phi_x\left(\omega = \frac{2\pi in}{d}\right),$$ ...
4
votes
0answers
36 views

Questions about point (xi) of consistency checks from Eigenvlaue - Eigenvector Identity by Terence Tao.

In the recent paper by Tao concerning the famous eigenvalue-eigenvector identity, I need some help in understand the (xi) point under basic consistency checks. The actual identity (2) given in the ...
2
votes
0answers
33 views

Characteristic function of non-central $\chi^2$

Consider $X_1,\ldots, X_n$ iid random variables with $X_1\sim\mathcal{N}(0,1)$, some real numbers $\alpha_1,\ldots,\alpha_n$. Define $Z:=\sum_{i=1}^n (X_i+\alpha_i)^2$ and $\kappa:=\sum_{i=1}^n\...
2
votes
0answers
58 views

What is the pdf of the distribution of the product of two normal random variable which does not follow $\mathcal{N}(0,1)$

I have two random variables which do not follow $\mathcal{N}(0,1)$. The characteristic function of the product of two random normal variables is $\frac{1}{\sqrt{\sigma^2 t^2 +1}}exp \left[ - \frac{\mu^...
3
votes
1answer
45 views

Given a r.v. $X$, can we choose a r.v. $Y$ independent of $X$ and with the same distribution?

The question originates in the following problem: Given that $\phi$ is a characteristic function, show that so is $| \phi |^2$. The solution given by a lecturer was as follows: Suppose $\phi$ is the ...
0
votes
0answers
56 views

Lévy-Khintchine canonical representaion of Gamma distribution

what is the Lévy-Khintchine representation of an infinitely divisible characteristic function(ch.f.). such as Gamma distribution (ch.f. $$f(x)=(1-\frac{ix}{\lambda})^r$$) I wonder if there is any good ...
2
votes
1answer
44 views

A subsequence of infinitely divisible characteristic function

I had to prove a proposition. Let $f$ be a characteristic function(ch.f.). There exists a sequence of ch.f. $\{\phi_{n_{k}}\}$ and $n_k$ is a sequence of positive integers tending to infinity. $f=(\...
1
vote
1answer
20 views

a locally constant function is decomposed into characteristic functions

Let $X$ be a Hausdorff topological space and $R$ a unital commutative ring with a discrete topology. Let $C_{R}(X)$ be the set of $R-$valued continuous function (i.e., locally constant) with compact ...
0
votes
1answer
23 views

Finding the characteristic function of $E[e^{iuXY}]$, given X, Y are standard normal, $Cov(X, Y) = 0$

I'm trying to understand the solution to this question: Suppose that $(X, Y)$ is a normally distributed random vector with $X \sim N(0, 1)$, $Y \sim N(0, 1)$, $\text{Cov}(X, Y) = 0$. Determine the ...
0
votes
0answers
18 views

If $X$ has Poisson distribution with exponential $\lambda$, for which $f$ is the characteristic function of $f(X)$ equal to $e^{\frac{φ_λ-λ(E)}n}$?

Let $d\in\mathbb N$ and $\lambda$ be a finite measure on $\mathcal B(\mathbb R^d)$. Remember that the characteristic function $\varphi_{\operatorname{CPoi}(\lambda)}$ of the compound Poisson ...
2
votes
1answer
43 views

Random walk with 3 values

I have the following question: Consider a one-dimensional random walk with step size $L$, where the probability of walking to the right is $1/4$, left is $1/4$, and staying where it is $1/2$. Suppose ...
0
votes
0answers
39 views

Does a characteristic function in the format $e^{v^{T}Kv}$ implies a random vector with a Gaussian distribution?

I know that if I have a Gaussian random vector $X$ with mean $m_X$=$[0]$, then I have a characteristic function $M_x(v)=e^{-\dfrac{1}{2}v^{T}Kv}$. In the other hand, if I have a characteristic ...
0
votes
0answers
34 views

What should $\mathbb{E}[\mathbb{1}_{A}(x)^{0}] $ evaluate to?

Please consider $\mathbb{E}[\mathbb{1}_{A}(x)^{n-k}]$. I am considering the expected value of an indicator to the power of $n-k$, particularly when $k=n$. I am concerned about the possibility of $\...
0
votes
0answers
32 views

Curve problem $x = x_0 + te^{x_0+1}.$

I'm studying characteristic curves in PDE. In one of the problems I was doing, I came up with the following characteristic equation: $$x = x_0 + te^{x_0+1}.$$ I need to isolate $x_0$ to proceed with ...
0
votes
0answers
20 views

Is there a PDF for a linear combination of Levy alpha-stable and normal distributions?

For independent random variables $X\sim\mathcal{S}(\alpha,0,1,0)$ and $Y\sim\mathcal{N}(0,1)$, what is the distribution of $Y\sigma_Y-X\sigma_X$ where $\sigma_X,\sigma_Y\in\mathbb{R}^+$ (such that $\...
0
votes
0answers
27 views

How to find priori information in Cramer Rao Lower Bound when the pdf of the parameter is unkown

In order to compute a Bayesian Cramer Rao Lower bound, we need to find the prior information of a random parameter $\theta$ which is a complex scalar parameter. The pdf of $\theta$ is $f(\cdot)$. ...
1
vote
1answer
28 views

The set of characteristic functions $\chi_x$ is a basis for $\mathcal{M}_{fin}(X,\mathbb{F})$

Let $X$ be a set and $\mathbb{F}$ a field. For $Y \subset X$, let $\chi_Y: X \to \mathbb{F}$ be the characteristic function of $Y$, that is the function defined by $$\chi_Y(x) = \begin{cases} 1 & ...
3
votes
2answers
89 views

Almost surely infinite random variable

I have three random variables, $X, Y$ and $Z$ that are related as follows. $$X = Y + Z$$ $X \sim Z$ whereas $Y$ and $Z$ are independent and $Y$ is std normal. $X\geq 0$ a.s. I want to show that $X = \...
0
votes
0answers
49 views

How can we show that $(\varphi_n^n(x))_{n\in\mathbb N}$ is convergent iff $(n(\varphi_n(x)-1))_{n\in\mathbb N}$ is convergent?

Let $d\in\mathbb N$ and $\varphi_n:\mathbb R^d\to\mathbb C$ be a continuous function$^1$ for $n\in\mathbb N$. How can we show that $(\varphi_n^n(x))_{n\in\mathbb N}$ is convergent for all $x\in\...
3
votes
2answers
68 views

Question about proof of a characteristic function

Let $X_k = 1$ with probability $0.5$ and $X_k = -1$ with probability 0.5, and let $X_k$ be independent random variables $k = (1,2,...,n)$. I was able to prove that the characteristic function $\...
1
vote
1answer
82 views

Continuous version of the Law of rare events

EDIT: Now I am just looking for a solution to the last part: Show a realisation of this problem such that a strong limit follows as well for $S_n$. In the simples version of the Law of rare events we ...
1
vote
1answer
98 views

Bound this complex integral in order to find the characteristic function

In the following problem, feel free to use the following facts for any complex number $z=x+iy\in\mathbb{C}$ with $\mathfrak{Re}z=x\in\mathbb{R}$ and $\mathfrak{Im}z=y\in\mathbb{R}$: $$ \begin{align*} \...

1
2 3 4 5
22