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
28 views

Characteristic Function and Convergence in Distribution of Sequence of R.V.

I am trying to solve the following: Let $X_1,X_2,...$ be a sequence of random variables with $P(X_n=\frac{k}{n})=\frac{1}{n}, k=0,1,2,...,n$. Find the characteristic function of $X_n$ and show that ...
2
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1answer
24 views

Find the elementary divisors of a matrix given its characteristic and minimal polynomials

This question comes from and old exam: Suppose the square rational matrix $A$ has characteristic and minimum polynomials $p_A(x) = x^6(x^2-2)^3(x^2+4)^2$ and $m_A(x) = x^2(x^2-2)(x^2+4)^2$ and $null A ...
3
votes
1answer
39 views

Computing Conditional Characteristic Function

I am trying to compute the characteristic function of the following: Let $X$ and $Y$ be random variables such that $Y\mid X = x\sim N(0, x)$ with $X\sim\mathrm{Po}(\lambda)$. Find the characteristic ...
3
votes
1answer
59 views

Show that $ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $ as $ \lambda \rightarrow \infty $

Given that the characteristic function for Y is $$ \varphi_Y (t) = e^{\lambda (e^{-t^2/2}-1)} $$ Show that $$ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $$ as $$ \lambda \rightarrow \infty $$ I've ...
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0answers
15 views

Show $\int_{-\infty}^{\infty}\,f(u,t)dG(u)$ is a ch.f. where $G$ is a d.f. ; $f(u,\cdot)$ is a ch.f. and $f(\cdot,t)$ is continuous.

Show $$\int_{-\infty}^{\infty}\,f(u,t)dG(u)$$ is a ch.f. where $G$ is a d.f. ; and $f(u,\cdot)$ is a ch.f. for each $u$ and $f(\cdot,t)$is coutinuous for each $t$. Note that ch.f. means ...
-3
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1answer
40 views

Characteristic function of an infinitely divisible distribution [closed]

I need to prove that for random variable $\xi$ which comes from infinitely divisible distribution characteristic function has no zeros, i.e. $\phi_{\xi}(u) \neq 0\: \: \forall \: \: u \in \mathbb{R}$ ...
0
votes
1answer
28 views

Recover the distribution of a Binomial random variable from its Characteristic Function

Hoping someone could show how to use the Characteristic Function of a binomial r.v. to recover its distribution. Using the inversion formula to recover the pdf of a r.v. with a continuous ...
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0answers
43 views

levy process and its characteristic function

Let $X(t)$ denote Levy Process. It can be proves that c.f of $X(t)$ is given: $E(e^{i\omega X(t)}) = e^{-\Phi(\omega)}$, where $ \Phi(\omega) = i \omega a - \int\limits^{-\infty}_{\infty} ...
1
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0answers
12 views

How to prove that the module of characteristic function is less than one

I would like to know if my resolution is right... I want prove than $|\varphi(t)| = \mathbb{E}[e^{i t X}]\leq 1$ , $\forall t \in \mathbb{R}$. $\it{proof:}$ First, note that $|\mathbb{E}[e^{i t ...
3
votes
1answer
61 views

Solving $u_t+u^2u_x=0$

I'm trying to solve the initial value problem with characteristis.: $$ u_t+u^2\cdot u_x=0\quad,\quad u(0,x)=f(x) $$ Where $u$ is a neat function with suitable requirements on its domain and its ...
0
votes
1answer
48 views

Why do characteristic functions use $e^{ix}$ and not $e^{-ix}$? Does it matter?

I've heard the characteristic function be described as the Fourier-Stieltjes Transform of the distribution measure of a r.v., but I was curious as to why it's written as $E[e^{ix}]$ and not the ...
2
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0answers
34 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
1
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1answer
24 views

probability density and distribution functions

I have $6$ independent and identically distributed variables such that $C_i \sim N(1000,400)$. 1) Calculate the density functions, distribution function and characteristic function of $C = ...
2
votes
1answer
39 views

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 $E\left[\mathbf{1}_{[a,b]}(Y)\right] = P(a \le Y \le b)$, ...
1
vote
1answer
25 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 ...
0
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1answer
31 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 ...
2
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0answers
31 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?
1
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1answer
51 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 ...
1
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1answer
21 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 ...
2
votes
2answers
41 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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0answers
26 views

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 ...
2
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0answers
40 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 ...
2
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0answers
31 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 ...
0
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0answers
43 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
43 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 ...
0
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1answer
23 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 ...
2
votes
1answer
39 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 ...
4
votes
1answer
91 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?
1
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1answer
29 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 ...
0
votes
1answer
48 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 ...
0
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1answer
16 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 ...
2
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1answer
43 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 ...
3
votes
2answers
35 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 ...
2
votes
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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0answers
25 views

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 ...
1
vote
1answer
38 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 $t$, ...
0
votes
1answer
21 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 ...
2
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0answers
29 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 ...
2
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1answer
31 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
25 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} ...
1
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1answer
38 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) ...
3
votes
1answer
137 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
48 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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0answers
33 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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0answers
22 views

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?
0
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1answer
114 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
2
votes
1answer
59 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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0answers
37 views

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
37 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 ...
1
vote
1answer
46 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 ...