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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Equation with mean of random variables

In a proof I found the following conversion $$E\left[|X|\mathbf{1}_{[a,b]}(Y)\right] = E\left[|X|P(a \le Y \le b)\right]$$ I understand, why $E\left[\mathbf{1}_{[a,b]}(Y)\right] = P(a \le Y \le b)$, ...
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1answer
21 views

Using characteristic functions to establish convergence

So I have found the Characteristic function of the variable $X_ \lambda$ to be: $$\psi_{X_\lambda}(t) = \psi_{b(\lambda)(Y_\lambda-\lambda)}(t)=\mathbf Ee^{itb(\lambda)(Y_\lambda-\lambda)}=\mathbf ...
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1answer
43 views

A reference for multi-dimensional characteristic functions

I'm looking for a well-written, rigorous and self-contained treatment of multidimensional characteristic functions, specifically Lévy's continuity theorem and the uniqueness theorem (which states that ...
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18 views

Determine if these are Characteristic functions

I'm having trouble determining if: $$\frac{sin(t^2)}{t^2}$$ and: $$\frac{1}{2} + \frac{1}{2}cos(t)$$ are Characteristic Functions. The first one I believe is very close to the sum of 2 uniformly ...
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1answer
25 views

Cauchy iid random variables and Strong Law of Large Numbers (helping understand)

Question: Let $(X_n)$ be a sequence of i.i.d. Cauchy random variables with density $\frac{1}{ π(1+x^2)}$. Use the characteristic function $φ(t) = e^{−|t|}$ of the Cauchy distribution to find the ...
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102 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}$ are i.i.d. $\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 ...
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21 views

Derivation of Gamma distribution characteristic function reference?

I was wondering if there was a derivation of the Gamma distribution characteristic function without expanding the $e^{itx}$ into an infinite summation?
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1answer
19 views

Differentiating Spitzer's identity

Let $(S_n)$ be an arbitrary random walk. Define $$M_n:= \max(0,S_1,...,S_n)$$ and $$S_n^{+} := \max\{0,S_n\}.$$ Spitzer's identity states that for $0<r<1$, we have $$\sum_{n=0}^{\infty} r^n ...
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1answer
34 views

Sequence of measurable functions $f_n=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$, uniform convergence

For each $n \in \mathbb N$, let $f_n:[0,\infty) \to \mathbb R: f_n(x)=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$. Show that there is no $E \subset [0,\infty)$ such that $|E|=0$ and $(f_n)_{n \geq 1}$ ...
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joint characteristic function of X and F(X)

X is a random variable. Its distribution function and characteristic function are $F_X$ and $\phi_X$, respectively. Then, we know, $F_X(X)$ follows uniform distribution. Let's say, $U=F_X(X)$. My ...
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29 views

Characteristic function of complex valued random variable

1) How is Characteristic function of a complex valued random variable defined? Should it be considered as vector of real random variables or the definition in wiki be used? 2) Also how is the ...
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23 views

How to use Duhamel's principle to solve wave equation

Let $x\in\mathbb{R}$,$t>0$, $$\frac{\partial u}{\partial t}+5\frac{\partial u}{\partial x}=e^{-t}sinx,$$ $$u(x,0)=(1-x^2)_{+}.$$ Using Duhamel's principle to solve it. By Duhamel's principle, the ...
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36 views

Differential equation whose solution is Erlang distribution

I am working on a proof (Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)) and got stuck. Now I would like try ...
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2answers
39 views

Solving characteristic equations…

Can someone explain to me what my teacher is doing? $x^2 - ax - b = 0$ ..? Isn;t he using the quadratic formula to solve this problem? If that's the case, then where is the $c$ at? Shouldn't he have ...
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1answer
29 views

Borel function and characteristic equation

Define a Borel probability measure $\mu_n $ by $\mu_n ({x}) = \frac{1}{n} $ for $x = 0, \frac{1}{n}, \frac{2}{n}, ..., 1-\frac{1}{n} $. Let $\eta$ be a Lebesgue measure on $[0,1]$. i) I'm to compute ...
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1answer
21 views

Characteristic function of an independent variable, does it involve complex values?

Let $$ x_k = \begin{cases} 1 & \mathrm{prob} (1/2)\\ -1 &\mathrm{prob} (1/2) \end{cases}$$ be independient random variables. Show that the characteristic function of the random variable ...
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1answer
26 views

Characteristic Function limit to 0

When calculating the limit of the following characteristic function $$ \frac{1}{n+1}\left[ \frac{1-\exp\left( \left(n+2 \right)it \right)}{ 1-\exp(it) } \right]$$ and taking its limit when ...
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1answer
293 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 ...
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1answer
18 views

Semi-Linear First Order PDE (with non-linear reaction term)

I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE: $\partial c \over \partial t$ + $K \over r^2$ $\partial c \over \partial r$ + $Da \ c \over ...
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1answer
32 views

show that $\exp(-t^4)$ is not a characteristic function

I found, by assuming that exists a random variable $X$ that accepts $\exp(-t^4)$ as characteristic function , $E[X^2] = 0$, which means that $P[X=0] = 1$ implying the function is $0$. It's acceptable ...
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1answer
85 views

Is there any short proof of this classical problem?

Let $X,Y$ be two i.i.d. r.v.'s with zero mean and unit variance. If $X+Y$ and $X-Y$ are independent, then $X$ and $Y$ are both standard normal distributed. Is there any short proof for this problem?
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1answer
33 views

Fourier transform of indicator function

Given a set of complex numbers $\mathcal A$, is there a convenient solution for the Fourier transform of its indicator function $\chi_{\mathcal A}(z)$? More specifically, if $\mathcal A$ is a set of ...
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1answer
15 views

Binomial-Poisson limit

I want to show that if $Z_n$ has the binomial distribution with parameters $n$ and $\lambda/n$ with $\lambda$ fixed, then $Z_n $ converges in distribution to the Poisson distribution, parameter ...
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1answer
37 views

calculating characteristic polynomial in $\mathbb{R}^n$

Given some hyperplane arrangement $\mathcal{A}$, we call any subset $\mathcal{B}\subseteq \mathcal{A}$ $\textit{central}$ if $$\displaystyle \bigcap_{H\in \mathcal{B}}H\neq \emptyset.$$ There is a ...
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2answers
33 views

How to use Cayley-Hamiltonian theorem in proving upper bound on linear space $W$?

If $W = span(I,A,A^1,A^2, \dots)$. What is the upper bound on dimension of $W$? All matrices are $n \times n$. I know that the dim($W$) $\leq n$, by the Cayley-Hamiltonian theorem. However, I don't ...
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1answer
23 views

Characteristic function of $\sum_{t=1}^N a_t X_t$ given certain independence conditions

Let $S=\sum_{t=1}^N a_tX_t$ where each $a_t$ is Bernoulli with probability $\frac{1}{2}$ for $1$ and also $\frac{1}{2}$ for $0$. Moreover, it is also given that the vector $(a_1,\ldots,a_N)$ is ...
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Characteristic function of an asymmetric Laplace distributed random variable

What is the characteristic function of a random variable with density $$f_X(x) = \frac{1}{2} [ 1_{x>0} \, a e^{-a x} + 1_{x<0} \, b e^{b x} ], \; \; \; \quad a,b > 0 \quad \quad ? $$ My ...
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1answer
349 views

$X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
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1answer
36 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, ...
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1answer
18 views

Independence of a random variable and a sub-$\sigma$-algebra

I am having trouble understanding one of the steps in the proof of the following lemma. Let $X$ be a random $d$-vector and $\mathcal{A}$ a sub-$\sigma$-algebra on the probability space ...
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27 views

Characteristics function and moments of multivariates

I have been reading this paper recently-- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.199.2157&rep=rep1&type=pdf This is a paper by Nengjiu Ju who uses talyor series to express ...
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1answer
29 views

New characteristic functions from old

I am doing an exercise which says: If $f$ is a characteristic function, then show that $$ F(t):= \int_0^{\infty} f(ut)e^{-u}du $$ is again a characteristic function. Is this answer correct? Let ...
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1answer
22 views

Proving $\chi_{f^{-1}(A)}(i) =(\chi_{A} \circ f)(i)$

Does this work for any $f$? \begin{equation} \chi_{f^{-1}(A)}(i) = \begin{cases} 1, & \text{if $i \in f^{-1}(A)$} \\ 0, & \text{if $i \not\in f^{-1}(A)$} \end{cases} = \begin{cases} ...
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1answer
133 views

If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?

The random variables take values in $\mathbb{R}^d$. I have tried to prove this using characteristic functions. Let $\hat{\mu}_{X_n},\hat{\mu}_{Y_n},\hat{\mu}_{Z_n}$ be the characteristic functions of ...
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1answer
35 views

Integrals of indicator functions question

I have a result $\int_X \int_Y \mathbb{1}[h(x,y) < \mu]dP(y)dP(x) < a$ and I am trying to resolve the integral $\int_X \int_Y \mathbb{1}[|f(x) - g(x)| > \frac{\mu}{2}] \mathbb{1}[h(x,y) ...
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45 views

Why does the characteristic function always exist?

I've read that the characteristic function of a probability distribution always exists because it's bounded. However, the characteristic function is still Taylor expanded in terms of the moments of a ...
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24 views

Characteristic function of basic affine process by rotation count algorithm

Hi everyone, I've had this frustating, silly problem for a while now. I've looked at the problem for a loooong time now, which may be one of the reasons I can't see the solution. I am trying to ...
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1answer
55 views

Odd Inverse of a Characteristic function

I saw this formula today: $$ \mathbb{P} \left[ X > K \right] = \frac{1}{2} + \frac{1}{\pi} \int_0^\infty Re\left( \frac{e^{-iuK}\varphi(u)}{iu} \right) du $$ Where $\varphi(u)$ is the ...
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1answer
52 views

How can you plot straight lines in Matlab using only values on x axis and the gradient of each line?

I'm trying to plot characteristics for the Burgers Equations. But I have to plot them only using the inbuilt function plot. It seems like a fairly straight forward problem but I still cant solve it
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Check if a distribution is discrete or continuous from the characteristic function of the distrution?

Is it possible to check if a distribution is discrete or continuous from the characteristic function/Laplace transform of a distribution?
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Some basic questions related to independence of random variables

I attend a lecture about Stochastic Processes even though I have not studied mathematics and some of the basics in probability theory are missing. So I hope you can help me with the following ...
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2answers
36 views

An integral related to Gaussian distribution

I am trying to evaluate the integral $$\int_{\mathbb{R}^d} e^{i \langle z,x \rangle} e^{- \langle x,A^{-1}x \rangle /2} dx $$ where $A$ is a positive-definite symmetric matrix. As a first step ...
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1answer
44 views

Inequality for Characteristic Function

For a discrete distribution the characteristic function $|\psi(u)|=1$ for other values of $u$ than $0$. We also know that $|\psi(u)|\leq 1$. How does this imply that for a continuous distribution we ...
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1answer
39 views

Recognize the distribution corresponding to this characteristic function

The characteristic function of a random variable $X$ is given as $$\phi_X(t)=\frac{3+\cos(t)+\cos(2t)}{5}; $$ what is the distribution of $X$? I was thinking of the discrete random variable $X=0,1,2$ ...
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1answer
25 views

Solving a characteristic equation of a differential equation of the 4th order.

My DE equation looks like this: $y^{(3)}+y^{(2)}+34y^{(1)}+40y=xe^{-4x}+2e^{-3x}cos(x)$ I'm having trouble solving for the characteristic equation $r^4+r^3+34r^2+40r=0$ I got it down to ...
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1answer
980 views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
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1answer
47 views

Solve $h(x)+h'(x)(8-x)-32=0$ for x.

Solve $h(x)+h'(x)(8-x)-32=0$ for $x$.Where $$h(x)=\frac{\frac{1}{16}x^2 - 2 x + 80}{\left(\frac{1}{16}x^2 - 2 x + 20\right)^2}$$ Should I go with characteristic equations? or is there another way. ...
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1answer
56 views

Can anything be learned about a probability distribution *directly* from its characteristic function?

Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us ...
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129 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 ...