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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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1answer
13 views

Show that stochastic integral w.r.t. Brownian motion is normal distributed

I want to show the following claim: Let $B$ be a one-dimensional Brownian motion and let $$I(\phi):=\int_0^1 \phi(s) \text{d}B_s.$$ Show that $\mathbb{E}(I(\phi))=0$ and $\mathbb{V}(I(\phi))=\int_0^1(...
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0answers
22 views

Show that $\frac{d}{dt} E\Big[\big |\phi_X(tZ)\big|^2\Big ]=2 E \left[ Z \phi_X(-t Z) \phi_X'(t Z) \right] $.

Let $Z$ be standard normal. Is the following sequence of steps correct? \begin{align} \frac{d}{dt} E \Big[ \big|\phi_X(tZ)\big|^2 \Big] &= E \left[ \frac{d}{dt} |\phi_X(tZ)|^2 \right] \\ &...
2
votes
0answers
36 views

Showing that the Distribution of a Random Variable Must be Standard Normal using Characteristic Functions [duplicate]

Question Let $X$ and $Y$ be i.i.d with means $0$ and variances $1$. Let $\phi(t)$ be their common characteristic function and suppose that $X+Y$ and $X-Y$ are independent. Show that $\phi(2t)=\phi(...
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2answers
29 views

Showing that the ratio of two standard independent normals is a Cauchy using Characteristic Functions

Question Let $X$ and $Y$ be independent standard normals. Use characteristic functions to find the distribution of $X/Y$. My attempt We will attempt to show that $Ee^{itX/Y}=e^{-|t|}$ (the c.f. ...
4
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1answer
85 views

Question involving Characteristic Functions and the Existence of a Distribution

Question Is it possible for $X$, $Y$ and $Z$ to have the same distribution and satisfy $X=U(Y+Z)$ where $U$ is uniform on $[0,1]$ and $Y$, $Z$ are independent of $U$ and of one another? The above ...
3
votes
1answer
31 views

If $\phi$ is a characteristic function, then $1-|\phi(2t)|\leq 8\{1-|\phi(t)|\}$

Question If $\phi$ is a characteristic function, show that $\text{Re}\{1-\phi(t)\}\geq \frac{1}{4}\text{Re}(1-\phi(2t))$ and deduce that $1-|\phi(2t)|\leq 8\{1-|\phi(t)|\}$. My attempt I have ...
1
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1answer
62 views

Show that $\int_{\mathbb{R}} |\mathbb{E}(e^{i \xi X})|^2 \, d\xi < \infty$ implies $P(X=x)=0, \forall x\in\mathbb{R}$. [duplicate]

Let $X$ be a real-valued random variable with $\varphi_X\in L^2(\mathbb{R})$, where $\varphi_X$ denotes the characteristic function of $X$. Prove that $P(X=x)=0, \forall x\in\mathbb{R}$. Since $\...
2
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0answers
36 views

Prove that $\phi(t)= \frac{1}{2} + \frac{e^{t^2}}{4e^{t^2} - 2}$ defines characteristic function

Let $\phi : \mathbb{R} \rightarrow \mathbb{R}$ be given by $$\phi(t) = \frac{1}{2} + \frac{e^{t^2}}{4e^{t^2} - 2}$$ Prove that $\phi$ is a characteristic function. My attempt: I know that there ...
2
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1answer
36 views

Prove that the given RV is normal

This is an old qualifying exam problem of probability theory. Let $X,Y$ be two iid random variables with mean zero and variance 1. Suppose that $X,Y,\frac{X+Y}{\sqrt{2}}$ are all identically ...
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2answers
38 views

Proof by characteristic functions that $X+Y$ and $2X$ are identically distributed

The exercise states that X,Y iid and we know that X+Y has Cauchy distribution. And they require to prove that 2X has also Cauchy distribution. Let me put it straight, I dont think I understand it ...
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0answers
39 views

Characteristic function equal to 1

Is the fact that the characteristic function of $X$ is equal to 1 enough to say that $X=0$ almost surely? Is this the correct proof? The characteristic function is determined uniquely. The random ...
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1answer
25 views

How to show Sub-independent Random Variables are uncorrelated.

I want to prove the following: If two RVs $X, Y$ are sub-independent, i.e., $\phi_{X+Y}(t) = \phi_X(t)\phi_Y(t), t\in\mathbb{R}$ then $X, Y$are uncorrelated. Keep $Cov(X,Y) = E(XY)-E(X)E(Y) = 0$ in ...
2
votes
0answers
47 views

If $\mathbb{E} e^{it(X+Y)} = \mathbb{E} e^{itX} \mathbb{E} e^{itY}$ then $X$ and $Y$ are uncorrelated

Let $X$ and $Y$ be random variables such that $\mathbb{E} e^{it(X+Y)} = \mathbb{E} e^{itX} \mathbb{E} e^{itY}$ and covariance exists. I want to show that they are uncorrelated, i.e., $\mathbb{E}(X -\...
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0answers
31 views

Characteristic function of infinitely divisible measure has no zero.

I want to show that the characteristic function of an infinitely divisible measure $P$ has no zero, directly by using the fact that for all $n\in\mathbb{N}$, there is a measure $P_n$, such that: $$\...
2
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0answers
27 views

Characteristic function and point masses

Let $X$ be some random variable with characteristic function $\hat{X}$. Show that $X$ has no point masses if and only if $$ \lim_{T\to +\infty}\frac1{2T}\int_{-T}^T\left\lvert \hat{X}(t)\right\...
0
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1answer
32 views

Functions of Characteristic Functions

Suppose that $\phi(t)$ is the characteristic function of a random variable. Show that the functions $\phi^2(t)$ and $|\phi(t)|^2$ are also characteristic functions. Attempt: So since $\phi(t)$ is a ...
1
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1answer
25 views

Why is it necessary X and Y beeing bounded when having same moments to be equal

Let $X,Y$ random variables in $[0,1]$ with $E(X^n)=E(Y^n) \,\forall n\in \mathbb N$. I want to show $X\overset{d}{=} Y$. $$E(e^{itX})=\int\sum \frac{(itx)^k}{k!}dF_X\overset{dom. conv.}{=}\sum\int\...
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1answer
13 views

Multivariable calculus discontinuities question

Question: If D is the set of discontinuities of $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, show that the set of discontinuities of $f_{\Phi_{A}}$ is contained in $D \bigcup \partial A$. So, $f_{\Phi_{...
1
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1answer
47 views

Fourier uniqueness theorem

I am reading Theorem 26.2 in the book "Probability and Measure" (2nd ed.) by Patrick Billingsley. In the proof of that theorem, the author claims that: Fact Let $\mu$ and $\nu$ be probability ...
2
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1answer
42 views

characteristic function and distribution completely determined by moments

Let $X$ be a real valued random variable. My textbook states that if the moment generating function $E[e^{sX}]$ is finite in a neighborhood of zero, the distribution of $X$ is determined completely by ...
2
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0answers
28 views

Continuity of a Characteristic function in real-analysis

Let $F \subseteq [0,1]$ be a closed set with empty interior such that $m(F)>0$ Prove that the characteristic function $χ_F$ denfined on $[0,1]$ satisfies the following property: for every set $A \...
1
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1answer
35 views

Poisson and convergence in distribution

Let $X_1,X_2,...$ be iid with mean $\mu$ and variance $\sigma^2$. Let $N_{\lambda}$ be poisson($\lambda$) independent of the $X_i$'s. (a) Find the limit in distribution as $\lambda \rightarrow \infty$...
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votes
1answer
22 views

characteristic function of a clopen set continuous?

Let $X$ be a compact Hausdorff space, $C(X)$ the continuous functions complex valued on $X$ and $U\subset X$ clopen. Is then the characteristic function $1_U\in C(X)$? $1_U:X\to\mathbb{C}$ is defined ...
0
votes
3answers
31 views

use indicator function to prove independence

I would like to know why for $\forall A \in \mathcal{F}$ $$E[1_{A}e^{itX}] = P(A)E[e^{itX}]$$ will imply the independence between $X$ and $\mathcal{F}$. Thanks in advance!
4
votes
1answer
52 views

Random Variable with Characteristic function $\frac{1}{2-\phi(t)}$

I am given that $X$ has c.f. $\phi(t)$, I need to find the random variable whose c.f. is equal to $\frac{1}{2-\phi(t)}$ in terms of $X$. My idea is that express $\frac{1}{2-\phi(t)}$ as a series, ...
3
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0answers
70 views

Proof of Convergence in Distribution for random variables with infinite variance

We are asked to prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{-3}$ outside $(-1,1)$, the following is true: $$ \frac{X_1+X_2 + \dots +X_n}{\sqrt{n\log n}} \xrightarrow{\...
1
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1answer
15 views

Finding a characteristic function of a product of two normal random variables [duplicate]

Let us define two, independent random variables $X, Y$. We know that: $$X - N(0, 1) \wedge Y-N(0,1).$$ Our task is to find characteristic function of a product of $X, Y$. I know that: $\varphi_X(t) = ...
2
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1answer
63 views

Does Cayley Hamilton Theorem apply for non-diagonalizable matrices as well?

Cayley Hamilton Theorem says that a matrix $A$ satisfies its characteristic equation. My professor proved this for diagonalizable matrices. What happens if $A$ is not diagonalizable? Does the C-H ...
1
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1answer
46 views

Characteristic function formula

Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and $\varphi_{X}$ the corresponding characteristic function. I try to show: \begin{equation} \mathbb{E}(X^2)=\...
1
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1answer
61 views

Characteristic function of ito process

I'm given process $X(t) = \int_{0}^{t} a(s)dB(s) $ where a(s) is a square-integrable deterministic function I need to show, that $E[e^{imX(t)}] = e^{-\frac{m^2}{2}\int_{0}^{t}a^2(s)ds}$ My attempt: ...
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1answer
17 views

Convergence of normal distributions with convergent mean and variance

Let $X_n$ be a sequence of random variables with $X_n \sim \mathscr{N}(m_k,\sigma^2_k)$. a) Assume that $m_k \rightarrow m$ and $\sigma^2_k \rightarrow \sigma^2 $. Show that $X_n \rightarrow^d \...
0
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1answer
32 views

Durrett's book Inversion Formula (Theorem 3.3.4)

I'm reading Durrett's probability book. In page 109, after stating inversion formula it gives a counter example for point mass function. I'm a little bit confused. The theorem is not true for all ...
3
votes
1answer
36 views

Mean of two i.i.d. random variables follows the same distribution, Cauchy distribution?

It is well known that if $X$ and $Y$ follow i.i.d. Cauchy distribution of scale $\gamma$, say $$ p_{\gamma} (x) = \frac{1}{ \pi \gamma ( 1 + x^2 / \gamma^2 ) }, $$ then their arithmetic mean $ ( X + Y ...
1
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2answers
55 views

Can the characteristic function of $\phi(t)$ satisfying $\phi(t)=\phi^2\left(\frac{t}{\sqrt{2}}\right)$ conclude it as C.F. of normal distribution?

Suppose $\phi(t)$ is a characteristic function of a random variable $X$. It satisfies $$\phi(t)=\phi^2\left(\frac{t}{\sqrt{2}}\right)$$ Further we know $\phi \in C^2(R)$. Can we derive $\phi(t)$ is a ...
1
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1answer
46 views

Characteristic functions total in $L^2$?

Prove that $\left\{\mathcal{X}(a,x):x \in [a,b]\right\}$ is total in $L^2([a, b]), dx)$ where $\mathcal{X}(c,d)$ is the characteristic function of $(c, d)$. What is the definition of being total? ...
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votes
1answer
40 views

Mean of zero mean random variables has Cauchy-Lorentz distribution under constraints on the characteristic function

Take $X_1 , X_2 , \cdots $ are $iid$ with zero mean. Take $$Z_n = \frac{X_1 + \cdots X_n}{\sqrt{n}} \stackrel{d}{\rightarrow} X$$ and $$Z_{2n} = \frac{X_1 + \cdots X_{2n}}{\sqrt{2n}} \stackrel{d}{\...
3
votes
2answers
66 views

With $X \sim Unif(0,1)$ what is the limit of $\frac{n}{x_1^{-1} + \cdots + x_n^{-1}}$

I am confused as to how I can tackle this question: With $X \sim Unif(0,1)$ what is the limit of $\frac{n}{x_1^{-1} + \cdots + x_n^{-1}}$. My assumption is that is $0$. but I would like to show that ...
0
votes
1answer
23 views

Show that $XY^{1/\alpha}$ is stable with index $\alpha \beta$

Suppose $X$ is symmetric and stable with index $\alpha$ and $Y$ is stable and nonnegative with index $\beta$. Show that $XY^{1/\alpha}$ is stable with index $\alpha \beta$. The text also gives the ...
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0answers
15 views

Showing equivalence of 3 statements

Let $X$ be a random variable in $\mathbb R^d$ whose distribution we denote by $P_X$ and characteristic function by $\phi_X$. How can I show that following statements are equivalent: 1.) $P_X=P_{-X}$ ...
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0answers
11 views

Bounding difference of characteristic functions in terms of total variation distance

I have a basic question regarding characteristic functions, which somehow I cannot figure out: Suppose that $X$ and $Y$ are two random vectors and $t$ is a fixed vector, all of the same dimension. ...
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0answers
20 views

A Question on Convergence in Probability of Conditional Characteristic Functions

This post is a bit long, so please don't be annoyed with me :). I have a question regarding convergence in probability of conditional characteristic functions, which I describe below: Let $X_n$ be a ...
0
votes
1answer
29 views

invertible matrix and determinant

Assume $(A + I_n)^m = 0$. Prove that $A$ is invertible and find $\det(A)$. I started by binomial expansion, and set it equal zero. is that correct? what would be the best approach?
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2answers
31 views

Question about Continuity of Characteristic Function

Is the function $f_n= \chi_{[\frac{j}{2^k} , \frac{j+1}{2^k}]}$, where $n=2^k+j$ with $0 \leq 2^k$, continuous on the interval $[0,1]$? I have no intuition regarding how this function looks ...
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0answers
40 views

Characteristic Function and Generalized Binomial Theorem

could you please help me on the following problem? I have the characteristic functions of two random variables $X$ and $Y$, denoted $\phi_X(u)$ and $\phi_Y(u)$, such that: \begin{equation*} \...
1
vote
0answers
17 views

Qualitative interpretation of PDF and its Characteristic function

tl;dr: I have limited information on the characteristic function of a PDF $W(x)$ I seek. I try to deduce a fitting qualitative picture of $W(x)$ from this information. I ask if you would agree with ...
0
votes
1answer
95 views

Convergence in distribution of sum of Bernoulli distributed random variables.

Let $X_i^{(n)} \sim \operatorname{Ber}(p_{i,n})$ for all $n\in \mathbb N$ and $i\in \{1,\dots n\}$ Bernoulli random variables on a probability space $(\Omega, F, \mathbb P)$, such that $X_1^{(n)},\...
0
votes
0answers
15 views

Transformation of Moments

Suppose that $X,Y$ are random-variables and that $f$ is an infinitely-differentiable function with infinitely differentiable inverse, such that $$ f(X)=Y. $$ Question How are the moments of $X$ ...
-1
votes
1answer
25 views

Check Characteristic function [closed]

How to check whether $\sin(t)\over t$ is a characteristic function or not? If $\sin(t)\over t$ is a distribution function.
3
votes
1answer
45 views

Need Help Interpreting the Sturm-Liouville Operator

I am given the following "Sturm-Liouville Problem with Operator $\mathcal{L}$ ": $$\mathcal{L}_{SL}=-\frac{1}{x}\left[\frac{d}{dx}\left(x\frac{d}{dx}\right)-\frac{1}{x}\right]$$ which is defined on ...
1
vote
1answer
63 views

Given a characteristic function, find the distribution.

I'm new to characteristic functions and I would really appriciate some help with the following question: "Give the distribution which has characteristic function $\varphi(t)=cos(t)$." I've tried to ...