Tagged Questions

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
30 views

uniform convergence of characteristic functions

Assume that a sequence of probability measures $\mu_n$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Prove that $\phi_n$ ...
0
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1answer
32 views

characteristic function differentiation

Let $\mu$ be a probability measure on $\mathbb{R}$. Then the characteristic function is: $$ \varphi: \mathbb{R} \rightarrow \mathbb{C} \;\;\ \varphi(t):=i\int_\mathbb{R} e^{itx}d\mu(x) $$ Prove with ...
2
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1answer
26 views

Imaginary solutions of a recurrence relation

How to solve this recurrence relation using characteristic equation and imaginary numbers? We have $a_0 = 0$ and $a_1 = 1$ , and for all $j\in\mathbb N$: $$a_{j+2} = 6a_{j+1} - 10a_j$$ I would ...
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3answers
45 views

Solve recurrence relation problem

This is a recursion problem that I am stuck at. I need to use the characteristic equation. Let $a_0, a_1, a_2, . . .$ be defined by $a_0 = 5, a_1 = 0$, and $a_{n+2} = a_{n+1} + 6a_n$ for $n \ge 0$. ...
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0answers
13 views

Characteristic function of truncated Cauchy distribution

Truncated Cauchy distribution (with a symmetric truncation $-X\le x\le X$) has the density $$f(x)=\frac{1}{2\arctan(X)}\frac{1}{1+x^2}$$ What is the characteristic function of this r.v.? Thank's! ...
0
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0answers
20 views

recognize the distribution corresponding to this characteristic function

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

Transform Characteristic Function to distribution Function

I was wondering how we can calculate the distribution function of this characteristic function, $$C(t)= \frac{3+\cos t + \cos2t}{3}$$
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0answers
45 views

How to use the characteristic function to obtain the first moment and find a meaningful bound for P(X>=k) based on its expectation only. [closed]

The probability generating function of the Poisson distribution with parameter $\lambda$ is given by $G(z) = e^{-\lambda(1-z)}$. Explain how to derive the characteristic function of the Poisson ...
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0answers
11 views

Order Statistics interval sizes

Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number ...
1
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1answer
32 views

Show that two sums have the same distribution?

I have not been able to show that the following two stochastic variables have the same distribution. My question is as follows: Let $$ X_1, X_2,..., X_n $$ be independent and identically ...
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2answers
37 views

On the characteristics function of smooth compactly supported distributions

My question is concerned with the Fourier transform of a density function of a continuous random variable (or characteristics function). In a book of Kim Lai Chung with the title "A course in ...
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0answers
20 views

By an example, show that..

Let $X_1$ ,$X_2$ be two r.v.'s with joint and marginal ch.f.'s $\phi_{X_1,X_2}$, $\phi_{X_1}$ and $\phi_{X_2}$. By an example, show that $\phi_{X_1,X_2} (t,t) = \phi_{X_1} (t) \phi_{X_2} (t)$ ...
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0answers
37 views

Clarification of Proof on Kac's Theorem for Characteristic Functions

There is a proof given here that I don't really understand, and was hoping someone more competent could explain it in some more detail: Moment generating functions/ Characteristic functions of $X,Y$ ...
2
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1answer
49 views

A linear combination of characteristic functions is a characteristic function?

Let $\phi_k(t)$ be the characteristic function of a random variable $X_k$, $k = 1,2,\dots$. Consider a set of positive real numbers $\{p_1, p_2, \dots \}$, take a function: $$\phi(t) = ...
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0answers
21 views

Recovering pmf from characteristic function

I'm having some trouble trying to recover the probability mass function of a discrete random variable from its characteristic function. I have seen that some continuous cases, you can recognize that ...
1
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1answer
70 views

Computing the characteristic function of a normal random vector

The characteristic function of a random vector $\boldsymbol{X}$ is $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}'\boldsymbol{X}}] $ Now suppose that $\boldsymbol{X} \in ...
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0answers
35 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
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2answers
75 views

How to show that $\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$

Let $x_n$ be a sequence of reals. Show that $$\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$$ Since the weak convergence is equivalent to pointwise convergence of characteristic functions ...
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2answers
62 views

Is $\exp(-2\sin^2t)$ a characteristic function?

Is $\exp(-2\sin^2t)$ the characteristic function of some random variable?
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1answer
38 views

Positive-definite + continuous at 0 $\Rightarrow$ continuous?

Let $F$ be a functional from $L_2(\mathbb{R})$ to $\mathbb{C}$ that is positive-definite*. We also know that $F$ is continuous at $0$. Can we deduce that $F$ is continuous over $L_2(\mathbb{R})$? ...
0
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0answers
41 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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2answers
36 views

characteristic function of $\sum_i^N X_i$, $N$ is a Poisson distribution

I have a series of $X_i$ random variables, identically and independent distributed. $S_n=\sum_i^N X_i$, with $N$ which has a Poisson distribution and is independent from $X_i$. I have to compute the ...
1
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1answer
21 views

Moment-determinacy in multivariate case

Let $X$ be a random vector with probability density $p$. In the scalar case I have learned that if the characteristic function of $X$ is real analytic, then all moments exist and $p$ is determined ...
2
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3answers
41 views

What is the set with characteristic function $\chi_A(x) + \chi_B(x)-\chi_A(x)\chi_B(x)$?

Suppose that $A$ and $B$ are subsets of $X$ Find the subset $C$ whose characteristic function is given by: $\chi_C(x)=\chi_A(x) + \chi_B(x)-\chi_A(x)\chi_B(x)$ The answer given is ...
5
votes
1answer
103 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
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0answers
92 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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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 ...
1
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1answer
46 views

Inequality on characteristic functions (probability theory)

Show that for every real characteristic function $\phi(t)$ we have $$1-\phi(2t) \le 4(1-\phi(t))$$ I am not sure where to begin. It seems I am missing some formula or theorem, or is it really that ...
3
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1answer
55 views

$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$

Hello everybody i need to show following equality $$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$$ Where $(X_i)_i$ are ...
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1answer
33 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
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0answers
32 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 ...
1
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1answer
66 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
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0answers
77 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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1answer
79 views

Prove if $E$ is a Lebesgue measurable set, there exists a continuous function $f$ differing from $\chi_{E}$ on a set of measure $< \epsilon$?

I am reviewing my analysis notes, and I don't really understand the proof given by my professor. He first proved if $E$ is a Lebesgue measurable set and $\epsilon > 0$, then there is an open set ...
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0answers
32 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
25 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 ...
0
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1answer
42 views

How to integrate $\int\mathbf 1_{(-\frac12,\frac12)}(z-w)\mathbf 1_{(-\frac12,\frac12)}(w)dw$

How to integrate $\displaystyle\int\mathbf 1_{(-\frac12,\frac12)}(z-w)\mathbf 1_{(-\frac12,\frac12)}(w)dw$ ? The integral should give a function of $z$, but I don't know how to compute. ...
0
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1answer
46 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
72 views

Show that Z cannot be turned into a vector space over any field. [duplicate]

Show that Z cannot be turned into a vector space over any field. So, we have 2 cases here. Case 1:lets suppose the charF=P, n does not equal 0, then (1+1+...+1)n=1n+1n+...+1n=n+n+...+n=pn=wchich ...
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1answer
27 views

Convergence of $\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$ for $u_n \to u$?

Let $u_n \to u$ in $L^2(\Omega)$ and let $I$ be an bounded interval. Does it follow that $$\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$$ at least for a subsequence of $u_{n_j}$ ...
0
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1answer
17 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
82 views

Characteristic Function Inversion

I am studying the relationship / bijection between characteristic functions and CDFs. In particular, given a characteristic function $\phi$ it is posible to recover the cumulative density function ...
1
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0answers
23 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?
5
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1answer
165 views

Continuous probability distribution with no first moment but the characteristic function is differentiable

I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere. Cauchy distributions do not ...
2
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0answers
70 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 ...
3
votes
1answer
99 views

Characteristic function under risk neutral measure

I am trying to derive a characteristic function (in Levy-Khintchine form) of a compound Poisson process $X_T$ under a risk neutral measure $\mathbb{Q}$, using the Esscher transfrom to change the ...
1
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4answers
129 views

Prove the matrix satisfies the equation $A^2 -4A-5I=0$ [closed]

How to prove that $$ A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} $$ satisfies the equation $A^2 -4A-5I=0$?
1
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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 ...
0
votes
1answer
68 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
votes
1answer
29 views

Empirical characteristic function

The ecf is $\phi_n(\omega) = \frac{1}{n}\sum_{j=1}^ne^{iX_j\omega}$. I'm stuck on trying to see why the following is true $$|\phi_n(\omega)|^2 = \phi_n(\omega)\phi_n(-\omega)$$ Wouldn't this imply ...