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
47 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
24 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
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1answer
32 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
34 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
122 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
108 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
147 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 ...
0
votes
1answer
19 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 ...
0
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1answer
21 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
99 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 ...
0
votes
1answer
59 views

Linear Transformations $T$ and $S$ and their Characteristic Polynomials

My friends and I cannot figure out this proof. We have part (a) done, but weren't not quite getting part(b). We think we need a change-of-basis equation. Any advice?
0
votes
1answer
101 views

Using characteristic function to deduce convergence of Bernoulli random variables

Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n = \sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in ...
4
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0answers
62 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 ...
4
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0answers
1k 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
556 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 ...
2
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0answers
80 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
170 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
54 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
340 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 ...
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0answers
13 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
63 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
34 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
62 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
175 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
185 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
113 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
400 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
155 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
16 views

Expected value and Differentiation of Characteristic function

Is there an example of random variable that has characteristic function to be differentiable at zero, but has no expected value?
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0answers
26 views

Inverting a difficult characteristic function.

Let $X$ be a random variable with characteristic function $$\phi_X(w)= \frac{1}{1-\frac{iw}{\lambda}e^{(\lambda-iw)\eta }}$$ where $\lambda$ and $\eta$ are constants. What is the pdf of $X$?
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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 ...
0
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0answers
33 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}] = ...
0
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0answers
29 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
46 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
43 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
34 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
42 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 ...
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0answers
67 views

Distribution infinitely divisible not stable

Give an example of a distribution that is infinitely divisible but not stable. There's a theorem that says: $X$ is a limit of a triangular array $\iff X$ is a infinitely divisible distribution. The ...
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0answers
39 views

A basic question on characteristic function

Suppose I have two random variables $X$ and $Y$ for which characteristic functions are same. Let $F$ and $G$ be their distribution functions. I have to prove that $F$ and $G$ have the same set of ...