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).
5
votes
2answers
240 views
Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$
Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below:
$\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$
Can it be proven ...
5
votes
1answer
217 views
For symmetric stable distributions, why is $\alpha \le 2$?
I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact.
Suppose we are trying to come up with stable distributions. From the definition, ...
5
votes
1answer
82 views
Determining if something is a characteristic function
Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$
Question: Given a function, say $\psi(t)$, how does ...
4
votes
3answers
430 views
Is it a characteristic function?
Can anyone explain, how can I prove either $\phi(x) = |\cos t|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.
4
votes
1answer
133 views
Combinations of i.i.d Inverse Chi-Square RVs and their characteristic functions
I am working on a few self-study problems in probability/measure theory and am stuck on characteristic functions. I have the following problem:
Given: $X_1,\ldots,X_n$ are iid inverse chi-square(1) ...
4
votes
0answers
251 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
452 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
votes
2answers
50 views
A criterion for independence based on Characteristic function
Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be ...
3
votes
1answer
128 views
Random variable with characteristic function $\large\frac{\phi(t)+\phi(-t)}{2}$ [duplicate]
Possible Duplicate:
Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$
If $\phi(t)$ is the characteristic function of a random variable $X$, then $\Re(\phi(t))$ is ...
3
votes
1answer
148 views
Primitive recursive functions and characteristic functions. Methods of proof- examples. Illumination.
I am puzzling over a sentence in an example in a textbook, showing that a function $f$, defined by cases, is primitive recursive.
Let $E$ be the set of even natural numbers. The function $f$ defined ...
2
votes
1answer
498 views
Characteristic function of a sum of Uniform random variables
Suppose I have $S=U_1+U_2+\dots+U_n$ where $U_i$ are distributed Uniform$[-1,1]$.
I am trying to show a couple of things. First, what is the characteristic function. I can show this easily enough for ...
2
votes
2answers
578 views
Characteristic function of exponential and geometric distributions
I'm trying to derive the characteristic function for exponential distribution and geometric distribution. Can you guide me on getting them?
Here is my solution so far:
Exponential Dist ...
2
votes
2answers
36 views
Evaluate $\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$
I'm trying to evaluate the integral $$\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$$
where $\chi_{[0,1]}(x)=1$ is the characteristic function, i.e. equals $1$ for $x \in ...
2
votes
2answers
45 views
Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite
There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$
R := \left(\limsup_{n\to\infty} ...
2
votes
1answer
75 views
Characteristic function converges pointwise
Is there any sequence of probability distributions such that their characteristic functions converge pointwise, but the sequence of prob. distributions itself does not converge weakly? :S
2
votes
2answers
69 views
Which distribution has the moment-generating function $\frac{\pi t}{\sin \pi t}$
The distribution $F(x) = e^{-e^{-x}}$ has moment-generating function $M_F(t) = \Gamma(1-t)$. From this it follows that the distribution of $X-Y$ for independently $F$-distributed $X,Y$ has the ...
2
votes
1answer
52 views
Missing assumption? (Convergence of random variables and characteristic functions)
Here are two exercises from my Probability book ("Probabilidade: um curso em nÃvel intermediário", by Barry James). (I have translated them from Portuguese to English and modified them a bit.)
...
2
votes
2answers
101 views
Obtaining cumulants using the characteristic function
If a random variable $x$ has a characteristic function $\phi(\omega)$, then the $n^{\mathrm{th}}$ moment of the distribution of $x$, $\mu_n$ can be calculated as:
$$\mu_n = ...
2
votes
1answer
72 views
Fourier transform of characteristic function in a sphere
A similar question was asked before for an interval in $\mathbb{R}$. I wonder how to do it for a characteristic function of $\{x\in\mathbb{R}^3:|x|<r\}$ i.e. I want to calculate $$ ...
2
votes
1answer
42 views
Characteristic function for positive part of random variables
I need your help in solving the following problem:
I have to calculate the characteristic function for the positive side of a random variable - simplest case: let $Y$ be standard normal and $Y^+ = max ...
2
votes
1answer
57 views
Show that the characteristic that passes through the point $(x,y)$ is given by $y(x)=\frac{1}{2}(x^{-2}-x_{0}^{-2})$
The function $u(x,y)$ satisfies the partial differential eqaution $x^{3}\frac{\partial u}{\partial x}-\frac{\partial u}{\partial y}=0$
Show that the characteristic that passes through the point ...
2
votes
1answer
146 views
Applying an inversion technique to Characteristic Functions
I am struggling with this concept (self-study). Could someone show me how to explicitly apply the inversion formula for these examples? I am working through about 15 examples, but these 3 seemed ...
2
votes
1answer
76 views
Characteristic functions (Statistics)
I would greatly appreciate any help with this problem. If $f_1, f_2 , f_3$ are three characteristic functions (in Statistics, e.g $E(\exp(itX)))$ such that $f_1*f_3=f_2*f_3$ for all $t$ and we are ...
2
votes
1answer
49 views
characteristic function properties
In lecture, we had the following corollary (without proof, unfortunately):
If $ A \in (0,2) $ and $X$ is a random variable (real-valued) with the following characteristic:
$$ \mathbb P(X > x) = ...
2
votes
0answers
30 views
Integral of the Normal Characteristic Function
The characteristic function of the $N$-variate Normal distribution is
$$\forall \mathbf{t} \in \mathbb{R}^N \quad
\psi(\mathbf{t}) \equiv \mathbb{E}\left(
e^{i\mathbf{t}X}\right) =
\exp \left( i{ ...
2
votes
0answers
58 views
Probability density function of A = B + C via Joint Characteristic function of B and C
This problem is actually a subproblem of a longer derivation that I am trying to understand. I hope that I striped away all the unnecessary stuff that is not relevant. Please correct me if the ...
2
votes
0answers
188 views
Distribution of the sum of iid Beta-Negative-Binomial random variables
I am facing a problem when trying to calculate the distribution of the sum of iid Beta-Negative-Binomial random variables or for that matter if only parameter $r$ is different.
To get a hint to how ...
1
vote
3answers
111 views
Is this function - characteristic function of a random variable?
$\phi(t)=
\begin{cases}
1,&\text{if $|x|< 1$;}\\
e^{-(|x|-1)^2} ,&\text{if $|x|\geq1$.}
\end{cases}
$
Can anyone help?
1
vote
2answers
210 views
Evaluation of Derivative Using $\epsilon−\delta$ Definition
Consider the function $f \colon\mathbb R \to\mathbb R$ defined by
$f(x)=
\begin{cases}
x^2\sin(1/x); & \text{if }x\ne 0, \\
0 & \text{if }x=0.
\end{cases}$
Use $\varepsilon$-$\delta$ ...
1
vote
1answer
186 views
Conditional characteristic function and independence
Suppose we have
$$E[\exp{(iu^{\operatorname{tr}}(X_t-X_s))}|\mathcal{G}_s]=\exp{\left(-\frac{1}{2}|u|^2(t-s)\right)}$$
for all $u\in \mathbb{R}^n$. Taking expectation leads to the conclusion that ...
1
vote
1answer
21 views
Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution
I am trying to derive Chi-square distribution. The random variale is
$$ U^2=\sum_{i=1}^k X_i^2 $$
where $X$ is a random variable with normal standard distribution.
What is the distribution of ...
1
vote
1answer
50 views
Proof by Characteristic Function
If $ X_1, X_2, \ldots,X_n$ are independent random variables variables with expectation $0$ and finite third moments.
Show that $$E((X_1+X_2+\cdots+X_n)^3) = EX_1^3+EX_2^3+\cdots+EX_n^3$$ using the aid ...
1
vote
2answers
35 views
Solution to ODE
I need to solve the following ODE
$\mu p_tF^\prime(p_t)+\frac{1}{2}\sigma^2p_t^2F^{\prime\prime}(p_t)-rF(p_t)+Ap_t+b=0$
I know a general solution is on the form
...
1
vote
1answer
62 views
Complex Conjugates of Characteristic Function
If $q(x)$ is the pdf, I can write it in terms of the characteristic function:
$$q(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-i\phi x} f_j(x,t,\phi)\, d\phi. $$
I see from the literature I'm ...
1
vote
2answers
78 views
Mean and deviation of characteristic function of an event
If I am given two events $A$ and $B$ and the probabilities $P(A), P(A|B), P(B), P(B|A)$, and am told that the random variables $X$ and $Y$ are defined as
$$X(a) = 1\ \text{if}\ a \in A\text{, else}\ ...
1
vote
1answer
204 views
relationship between non-central moments and the characteristic function of a distribution?
Given a random variable $X$, consider the k-th (non-central) moment about $a$, $E \left[ ( X - a )^k \right]$
Is there any relation of this value to the chracteristic function of variable's ...
1
vote
1answer
46 views
For any $c_1,c_2\in\mathbb{C}, E(c_1Z+c_2)=c_1E(Z)+c_2$
When dealing with real-valued RVs, the extensions of expectation and variance are quite clear to me. For example, showing
$E(aX+b)=aE(X)+b$ and $var(aX+b)=a^2var(X)$ is relatively straightforward to ...
1
vote
1answer
96 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 ...
1
vote
0answers
68 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 ...
1
vote
0answers
40 views
Plotting characteristic equations maple
I'm trying to plot the characteristics for the following PDE with initial conditions:
$$u_t +uu_x =0$$
where
$$u(x,0)=a \ for\ x<-1, \ b\ for\ -1<x<1, \ c \ for\ x>1.$$
I'm first ...
1
vote
1answer
66 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 ...
1
vote
0answers
115 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 ...
1
vote
2answers
163 views
Solve PDE using method of characteristics
The following problem I find challeging. Can you help me find a solution? The question is as follows:
Determine the solution (in explicit form), using the method of characteristics, of the ...
1
vote
0answers
112 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 ...
1
vote
1answer
58 views
Deriving the characteristic function for $N(0,2)$
Could someone please help me with an easy derivation of the characteristic function for a $N(0,2)$ distribution? Or a link to somewhere it is done.
0
votes
2answers
32 views
Linear Algebra - Elementary Canonical Forms
Let $N$ be a $2\times2$ complex matrix such that $N^2=0$. Prove that either $N=0$ or $N$ is similar over $\mathbb{C}$ to
$$\begin{pmatrix}0&0\\1&0\end{pmatrix}$$
0
votes
1answer
29 views
Property of the characteristic function for events
We got to show the following equality:
$1_{A_1 \cup...\cup A_n} = 1 - \prod \limits_{i=1}^n (1 - 1_{A_i})$
First I would like to ask for hints for how to proove this equation (no solution though, I ...
0
votes
1answer
52 views
Integral of characteristic and density function
Let X and Y be random variables (real valued) with density functions $f_X, f_Y$ and characteristic functions $I_X, I_Y$.
How can we show that:
$ \int_{-\infty}^{\infty}{I_X(y) f_Y(y) e^{-ity}dy} = ...
0
votes
1answer
56 views
Characteristic Function and Expected Value
I have another question regarding the Indicator Function, namely understanding the following equality:
$ 1 - E[\prod \limits_{i=1}^n (1 - 1_{A_i})] = \sum_{k=1}^{n} (-1)^{k+1} \sum_{1 \leq i_1 < ...
0
votes
1answer
120 views
The second order approximation of the Taylor expansion of Characteristic functions:
Let $X$ be a random variable with continuous density $\rho(x)$. Assume that $X$ is symmetric and $\vert X\vert<L$. Since it has a bounded support, all moments of $X$ are well-defined. Let $m_i$ ...
