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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3
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1answer
53 views

the density of the sum of $n$ random variables with uniform distribution on $(-1,1)$

Let $X_n$ be an iid sequence of random variable with uniform distribution on $(-1,1)$. Using characteristic functions prove that $X_1+X_2+...+X_n$ has density $$f(x)= \frac{1}{\pi} ...
1
vote
1answer
26 views

PDE $ u_{x}+u_{t}+f(x)*u=0$

How would I solve this pde using characteristic line? $u_{x}+u_{t}+f(x)u=0$---arbitrary function f $u(x,0)=u_{0}(x)$---$u_{0}$ can be any value $u(0,t)=\varphi(t)$---non-homogeneous where $u(x,t)\ge ...
1
vote
1answer
35 views

Why can I use the Riemann-integral here?

Let $Z\sim\mathcal{N}(0,1)$ (i.e. a random variable which distribution is the standard normal distribution). Determine the characteristical function of $Z$. It is $\mathbb{P}_Z=f\lambda$ ...
1
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1answer
39 views

characteristic function characterize the distribution

Theorem: Let $\phi(t)=\int{e^{itX}dF_X}$ be a characteristic function of a random variable $X$. Then $\displaystyle \lim_{T \to \infty}\int_{-T}^{T}{{\frac{e^{-ita}-e^{-itb}}{it}}\phi(t)dt}=P(X\in ...
1
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1answer
182 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
1
vote
1answer
144 views

Limit of characteristic functions

Let $\xi_1 ... , \xi_n$ be iid with $E \xi_i^2 < \infty $ what is $$ \lim _{n\rightarrow \infty} \varphi_{\bar{\xi}}$$ where $\varphi$ is the caracteristic function and $\bar{\xi}$ the mean of all ...
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1answer
111 views

Calculation of the characteristic function

We've given: $(X_n)$ ($n \ge 1$) a sequence of iid random variables such that $$ \mathbb P(X_1 = 1) = \mathbb P(X_1 = -1) = \frac{1}{2} \text{ and } L_n = \sum_{k=1}^n k^{\frac{-1}{2}} X_k. $$ What ...
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1answer
165 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
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1answer
39 views

Characteristic function of logarithm of random variable

If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$? I know that $\phi_X(t)=E[e^{itX}]$ and ...
0
votes
1answer
16 views

characteristic equation in pde

In the PDE: $ yU_y-xU_x=1$ how did the characteristics become $dx\over -x$=$dy \over y$ =$du \over 1$.Can someone please expalin how these charactristic equations were obtained
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1answer
49 views

Calculation of characteristic functions of Levy processes

Let us say we have some Levy process $X_t$ and want to calculate its characteristic function, $E[e^{iuX_t}]$ for a certain value $u$. Is there a general procedure for this? I can imagine a way of ...
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1answer
22 views

Find the standard representation of a function and the Lebesgue integral

Find the standard representation of the function f defined by f(x)=[x] for −1≤x≤3, f(x)=0 otherwise. determine the integral R of fdu I came across this question while studying and began to attempt ...
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1answer
159 views

Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal ...
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0answers
2k views

Characteristic functions of random variables (Poisson, Gamma, etc.)

My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is ...
4
votes
0answers
587 views

Characteristic functions based proof problem.

I am trying to show that if $T$ be a closed bounded interval and $E$ a measurable subset of $T$. Let $\epsilon >0$, then there is a step function $h$ on $T$ and a measurable subset $F$ of $T$ for ...
3
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0answers
78 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows that ...
3
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0answers
386 views

Distribution of the sum of iid Beta-Negative-Binomial random variables

I am facing a problem when trying to calculate the distribution of the sum of iid Beta-Negative-Binomial random variables or for that matter if only parameter $r$ is different. To get a hint to how ...
2
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0answers
49 views

Characteristic function of compound Poisson process

It is widely known that the characteristic function of a compound Poisson process is $$ \phi_X(u) = \exp \left(t\lambda \int_{\mathbb{R}} (e^{iux}-1) F(dx) \right). $$ But if I try to derive it via ...
2
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0answers
87 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
2
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0answers
211 views

Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
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0answers
69 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ...
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0answers
68 views

properties of characteristic function

Let $X,Y$ be two independent random variables having the same distribution, centred and with variance 1, $\phi$ is the characteristic function of $X$ and $Y$. If $X+Y$ and $X-Y$ are independent, show ...
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65 views

Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y $ have the following joint ...
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0answers
29 views

Integrating characteristic functions.

I need to find the radon transform of the following function. But I got stuck in finding this integral. Let $\chi$ be given by $$\chi(t) = \begin{cases} 1 & |t|< 1/2 \\ 0 & ...
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0answers
24 views

$X\sim\mathcal N(0,1)$, Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd?

If $X\sim\mathcal N(0,1)$ Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd ? ($\Phi_X^{(j)}(0):j^{th}$ derivative of the characteristic function of the r.v. $X$) We computed ...
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0answers
21 views

Finding Characteristic Exponents for $x^2 (x-1)^2 y'' + 4 (x-1)y' - 4x^2 y = 0$

I've found that the only regular singular point of this differential equation: $$x^2 (x-1)^2 y'' + 4 (x-1)y' - 4x^2 y = 0$$ is $x = 1$. How do I determine the characteristic exponents for it?
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0answers
18 views

Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
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0answers
26 views

Show that if X has a density f such that f’ exists and is integrable?

Show that if $X$ has a density $f$ such that $f'$ exists and is integrable, then its characteristic function has the property : $\phi(t)=ο(t^{-1} )$ as $t\to \infty$. Hint: If $X$ has a density ...
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0answers
70 views

Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see ...
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0answers
38 views

I want to show $\phi_{X}(a_{1},a_{2},\cdots,a_{n})=\prod_{i=1}^{n}\phi_{X_{i}}(a_{i})$

Let $n \in \mathbb N$ and $X$ be an $\mathbb R^n$ valued random variable on $(\Omega ,\mathcal F,P)$ Define its characteristic function to be $$\phi_{X}(a)=E(e^{i\langle X,a\rangle})$$ where $a \in ...
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0answers
69 views

Characteristic function of an r.v. with finite variance and zero mean.

Suppose $E{|X|^{2}}<\infty$ and $E{X}=0$. Show that Var(X)$= \sigma^{2}<\infty$ (done), and that $\varphi_{X}(u)=1-\frac{1}{2}u^{2}\sigma^{2}+o(u^{2})$ (what I can't figure out how to find, ...
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0answers
214 views

Characteristic Function of a Double Exponential (Laplace) Distribution

Let X have the double exponential (or Laplace) distribution with $\alpha =0$, $\beta = 1$: $f_{X}(x)=\frac{1}{2}e^{-|x|}$, $-\infty < x < \infty$. Show that $\varphi _{X}(u)=\frac{1}{1+u^{2}}$. ...
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0answers
225 views

Characteristic functions in set theory

The book I am studying has the definition of a characteristic function as follows. Let $A\subseteq{X}$. Then $$\chi_A(x) = \begin{cases} 1, & \text{if $x\in{A}$} \\ 0, & \text{if ...
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0answers
73 views

Series expansion at $0$

Given: $X$, $Y$ iid random variables, $\mathbb E(X) = 0$, $\mathbb E(X^2) = 1$; $X+Y$ and $X-Y$ are independent; $\phi$ is the characteristic function of $X$ and $Y$ and $ \psi: t \rightarrow ...
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0answers
114 views

Probability density function of A = B + C via Joint Characteristic function of B and C

This problem is actually a subproblem of a longer derivation that I am trying to understand. I hope that I striped away all the unnecessary stuff that is not relevant. Please correct me if the ...
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0answers
499 views

Continuity of a Characteristic function

Let $A$ be a subset of $\mathbb{R}^n$. Show that the characteristic function $\chi_A$ is continuous on the interior of $A$ and on $A^c$ but discontinuous on the boundary of $A$. My attempt: Suppose ...
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0answers
161 views

Questions about characteristic functions and continuity

If there is a subset $S$ in $\mathbb{R}^n$ consider the characteristic function $\chi_S: \mathbb{R}^n \to \mathbb{R}$. i) what would be the value of $\chi_S(p)$ for any $p \in \mathbb{R}^n$? Attempt ...
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0answers
37 views

Integral of Normal Distribution with imaginary unit

Hi I need some help with the following integral. $$ \int_{-\infty}^{\infty} \operatorname{e}^{itx} \cdot \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \operatorname{e}^{\frac{-(x - \mu)^2}{2\sigma^2}} \mathrm ...
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0answers
13 views

characteristic function of a stochastic process with stationary and independent increments

Let $(X_t)_{t\geq 0}$ be a stochastic process with independent and stationary increments. I have to show that $E[e^{itX_1}]=\phi^n(t)$ Since increments are independent, I can write ...
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0answers
15 views

Characteristic Function of a Spextral Density Function

I am struggling with understanding the link between the Spectral Density Function and the Characteristic Function. In particular, can you find the Characteristic function when only the SPX and ...
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0answers
31 views

Measure of $\chi_\mathbb{Q}(x)$?

$\chi_\mathbb{Q}(x) = 1$ if $x \in \mathbb{Q}, 0$ otherwise. Well $\chi_\mathbb{Q}(x)$ is a measurable function if $\mathbb{Q}$ is a measurable set. $\mathbb{Q}$ is a measuable set under the Borel ...
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0answers
16 views

Charateristic function evaluation

I have a signal given by the following equation: $y_k = X_k S_0 + \sum_{l=0 \& l\neq k}^{N-1}S_{l-k}+n_k$ where $X_k$ are independent and identically distributed random variables. $n_k$ is a ...
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0answers
42 views

Characteristic function of a exponential random variable, problems with complex integral.

I tried to compute the characteristic function of a random variable, which is exponential distributed with parameter $\lambda$: \begin{align*} \varphi(t) &= \mathbb E[e^{itX}] = ...
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0answers
38 views

characterization of characteristic functions (Bochner Theorem Proof?) Simple case.

Prove the following theorem: Let $\phi: \Bbb R \to \Bbb C$. $\phi$ is the characteristic function of a real random variable $X:\Omega \to \Bbb R$ if and only if $\phi(0)=1$ $\phi$ is uniformly ...
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0answers
21 views

Proving the characteristic equation

Consider the recurrence relation: $a_n = \alpha_1 a_{n-1}+\alpha_2 a_{n-2}+...+\alpha_k a_{n-k} ,$ where $\alpha_1 , \alpha_2 , ... \alpha_n $ are constants. 1) Prove that if $b$ is a non-zero ...
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0answers
51 views

Possible values of characteristic functions (Fourier transforms)

Can a characteristic function $\varphi_{X}(u)$ from probability theory (the Fourier transform of a probability measure) ever equal zero for either any value of $x$ or any value of $u$? This has been ...
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0answers
44 views

For X, Y real valued and independent, and X and X+Y having the same distribution, Y=0 a.s.

Let X, Y be real valued and independent. Suppose X and X+Y have the same distribution. Show that Y is a constant r.v. equal to 0 almost surely. Here's what I have so far: By the uniqueness of ...
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0answers
38 views

Limit of the expectation of the sum

Show that for $g(t)= E \left\{\sum_{n=3}^{\infty}\frac{(iut)^{n}}{n!}\right\}$ that $\lim_{t \to 0} \frac{|g(t)|}{t} =0$. I think I should bound it and then use LDCT, but I'm having trouble doing ...
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0answers
43 views

Find the characteristic equation of a recursive function

I want to determine whether the following recursive function is unstable; $$ x(t+1) = \left( wx+sx(t)^b \over w+x(t)^bs + (1-x(t))^b(d-s) \right) $$ Wikipedia is telling me that I want to have the ...
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0answers
13 views

multi-normal N(m,C): if C is not invertible, what's the impact?

I'm reading Bernt Oksendal's "Stochasticc Differential Equations" and got confused at the definition of multi-normal distribution. In Appendix A, page 307 (sixth edition) it says: a random variable ...