Questions tagged [characteristic-functions]
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.
1,227
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Sufficient condition for the characteristic function of a discrete distribution to be decreasing
Let $X$ be a discrete random variable defined on the lattice such that $P(X \in \mathbb{Z}/h) = 1$, where $\mathbb{Z}/h$ represents the set of integers scaled by a factor of $h$. I wonder if there ...
- 77
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42
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characteristic function in definition of a probability density function
I came across this function in a paper (used to compute the distribution of some real-valued random variable $\lambda_{c}$):
$$\large f({\lambda})=\sum\limits_{c \in C_{S}} \frac{\chi_{\{\lambda\}}(\...
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Is it true differentiability of characteristic function at 0 implies existence of the first moment.
The context is the following:
Prove $|\cos(t)|$ is not a characteristic function of a distribution, while $\cos(t)$ is a characteristic function.
I know you can prove it by inverse Fourier ...
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Quotient of scaled independent Erlang variables is F-distributed
Let $ X_j \sim Erlang(a_j,b_j)$ for $ j=1,2$ be independent random variables. Show that $ \frac{a_2 b_2 X_1}{a_1 b_1 X_2 } \sim F_{2b_1,2b_2}$
My idea was first to write $\frac{a_2 b_2 X_1}{a_1 b_1 ...
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31
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Difference between characteristic functions of two random variables in terms of expection of squared difference
Let $U_1, \ldots, U_n$ be uncorrelated random variables with zero mean and finite variance. Define $X = \sum_{i=1}^n U_i$ and $X_m = \sum_{i=1}^m U_i$ for $m \le n$. Let $\varphi_X(t)$ and $\varphi_m(...
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Compute the characteristic function of $X\sim\text{Poisson}(Z)$ where $Z$ is exponentially distributed.
The question comes from this post Poisson Process with Randomly Distributed Time.
Let $X\sim\text{Poisson}(Z)$ where $Z$ is exponentially distributed. Suppose that $X$ and $Z$ share the same parameter ...
- 5,585
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1
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Inverse the mapping from a subset $B$ of $X$ to its characteristic function $\chi_B: X \to \{0,1\}$
I am trying to find that the the power set P(X) of a set X is isomorphic to the set of maps from X to {0,1}
That map should be given by the mapping of a subset $B$ of $X$ to its characteristic ...
- 469
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42
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Generate sample values of a distribution with known Characteristic Function in R
I have given the characteristic function
$$\mathcal F(p)(x)=\frac{\exp(-|x|^s)}{(1 + x^2)^5}$$
where $s=0.5, s=1, s=1.5$ or $s=2$.
Now I want to get sample values of the distribution belonging to the ...
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For random variable X on a probability space there is another probability space with random variable Y that has the same probability as its negative
$\newcommand{\P}{{\mathbb{P}}} \newcommand{\A}{{\mathcal{A}}} \newcommand{\O}{{\Omega}}$
$\newcommand{\Pp}{{\mathbb{P}^\prime}} \newcommand{\Aa}{{\mathcal{A}^\prime}} \newcommand{\Oo}{{\Omega^\prime}}$...
- 21
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How to prove that the characteristic function of $n \bar{X}^2$ converges to that of chi square
Let $X_i$ be iid random variables with $E[X_1] = 0$ and $E[X_1^2] = \sigma^2$. I wonder how to show that the characteristic function of $n \bar{X}^2$ converges to that of $\sigma^2 Y$, where $Y$ is a ...
- 1,240
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Explicit unique solution formula for linear constant coefficient recurrence relation depending on a parameter when eigenvalues change multiplicity
Consider the three-term recurrence relation on $y^{n}=y^{n}(\theta) \in \mathbb{C}$ depending on the parameter $\theta$:
$$
y^{n+1}(\theta) = \alpha_{2}(\theta) y^{n}(\theta) + \alpha_{1}(\theta) y^{n-...
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Showing weak convergence of dirac measure using characteristic functions [closed]
I want to show if $\mu_n=\frac{1}{n}\delta_n+(1-\frac{1}{n})\delta_0$ converges weakly using characteristic functions. I know that the characteristic function of $\delta_0$ is the constant function $1$...
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Inversion formula for characteristic functions whose argument is complex with non-zero real part
In this paper they use an inversion formula for characteristic functions whose argument is complex with non-zero real part. Namely, given a random variable $X$ taking values in $(0,\infty)$, they ...
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PDF or CDF of product of two independant random variables
I'm looking for a formula for computing the PDF (I would prefer the CDF if possible) of the product of two independent random variables.
I found the Mellin integral/transform is the analytical ...
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2
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Why is $\lim_{n\to \infty} n\chi_{(0,\frac{1}{n})} = 0$? [closed]
Why is $\lim_{n\to \infty} n\chi_{(0,\frac{1}{n})} = 0$?
Can you provide a short (trivial) explanation?
The sequence in the limit only have two possible values: $0$ and $n (\to \infty)$ so it is a ...
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1
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Can $\mathbb{R}$ be expressed as a disjoint union of two dense locally non-zero measure measurable subsets?
I was trying to prove that piecewise-constant functions on $\mathbb{R}$ are not dense in $L^\infty(\mathbb{R})$ and I was looking for a $L^\infty$ function that cannot be "approximated" with ...
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Exercise 2.3.29 from Stroock's "Probability Theory: an Analytic View"
I am working on Exercise 2.3.29 from Stroock's "Probability Theory: an Analytic View".
Let $X$ be a multivariate normal random variable defined on a probability space $(\Omega, F, P)$ with ...
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Condition guaranteeing integrability
Let us consider a Levy process $X_t$ with the characteristic function of the form:
$$
\hat{\mu}(t) = \text{exp}\biggl( i\gamma t - \frac{\sigma^2}{2} t^2 + \int_{\mathbb{R}} (e^{itx} - 1 - itx \mathbb{...
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Recurrence of an irreducible random walk (Chung - Fuchs Theorem)
I am studying the Theorem by Chung and Fuchs (1951) from the book Probability Theory, 3rd version, by A. Klenke and I have a couple of steps for which I need some clarification.
The theorem states ...
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Characteristic Function of $\mathbb{N}$-valued random variable independent of sequence of random Variables
I'm currently studying for an exam on probability theory and encountered a problem on which I got quite stuck. It goes as follows:
Let $(X_n)_{n \in \mathbb{N}}$ be an independent family of random ...
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216
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Bound on the absolute value of the product of characteristic function and its derivative in terms of variance
Suppose a random variable $X$ such that $E[X] = 0$ and variance $0 <\sigma^2 < \infty$ with characteristic function $\varphi(t)$. I founded the upper bound $\left|\varphi(t) \frac{d\varphi(t)}{...
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114
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Proof of the Bochner-Minlos theorem for the Schwartz space
I am currently reading a paper about fractional Gaussian fields and try to find a proof of the Bochner-Minlos theorem for the Schwartz space. The version I consider is the following:
A complex valued ...
- 171
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1
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61
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Characteristic function on I is uncountable
I studied characteristic function in real analysis in Richard R goldberg book for methods of real analysis and there is an exercise problem asking to prive that characteristic function on I is ...
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2
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Show that $\lim_{x\rightarrow0}\frac{e^{\iota x}-1}{\iota x}=1$
(a) Show that $|e^{\iota a}-e^{\iota b}|\le\min\{2,|a-b|\}$ for all $a,b\in\mathbb{R}$. Hint. Write it as an integral from $a$ to $b$.
(b) Show that $$\lim_{x\rightarrow0}\frac{e^{\iota x}-1}{\iota x}=...
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Find parameters for process to become gaussian
Find parameters $ a,b,c $ for process $ aW_t^2+cW_{bt^2+a} $ to make it gaussian.
The process is gaussian if
$$ E \left [ \exp \left ( i \sum_{l=1}^{k} s_l Y_{t_l} \right ) \right ] =
\exp \left ( -\...
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1
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Probability density of two random variables using characteristic function
I've been trying to solve the following question :
$X$ and $Y$ are two real random variables with a probability density of :
$$f(x,y) = e^{-y} *\mathscr{1}_{0<x<y}(x,y)$$
where $\mathscr{1}$ is ...
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What if the initial data curve coincides with a characteristic?
I am trying to solve this problem:
$u_x+2xu_y$=y, with u(0,y) = 1+$y^2$ for $\frac{-1}{2} < y < \frac{1}{2}$ and determine the greatest region in which a unique solution exists.
I have found the ...
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Compute $\lim_{n \to \infty} \mathbb{P}(|\sum_{k=1}^{n} X_{k}^{-1}|> \pi n/2)$
Question: Let $(X_k)_{k \geq 1}$ be a sequence of independent random variables with uniform distribution on $[-1,1]$ Compute $\lim_{n \to \infty} \mathbb{P}(|\sum_{k=1}^{n} X_{k}^{-1}|> \pi n/2)$
I ...
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Characteristic function of Gamma-OU process (cross-posted from quant.stackexchange)
(cross-posted from quant.stackexchange at https://quant.stackexchange.com/questions/75702/characteristic-function-of-gamma-ou-process)
Consider the Gamma-Ornstein-Uhlenbeck process defined in the way ...
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Uniqueness of Compound Poisson Distribution
Let $X$ be a random variable such that $X =^d \sum_{i=1}^N Y_i$ where $N$ is Poisson and $(Y_i)$ is a sequence of IID random variables independent of $N$. Then $X$ is said to have a compound Poisson ...
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Let $X$ be a normal standard random variable and $a>0$ compute the characteristic function of $ Y:= X 1_{|X|\leq a}-X 1_{|X|>a}$
Let $X$ be a normal standard random variable and $a>0$ compute the characteristic function of
$$ Y:= X 1_{|X|\leq a}-X 1_{|X|>a}$$
For definition
$$\varphi_Y (t)=E[e^{itY}]= \int_{-\infty}^{\...
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Finding the probability density function of a random variable using characteristic function.
Question
Let X be a random variable with characteristic function $\varphi$. Show that
$$
\mathbb{P}(X = a ) = \lim_{T \rightarrow \infty} \frac{1}{2T} \int_{-T}^T e^{-ita}\varphi(t)dt.
$$
Attempt
Let $...
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How to show that $\mathbb{E} [ e^{itX} ] = 1 \implies e^{itX} \equiv 1$ a.s.?
Let $X : \Omega \to \mathbb{R}$ be a random variable, and consider its characteristic function:
$$\phi_X(t) = \mathbb{E}[e^{itX}]$$
Suppose that $\phi_X(u) = 1$. How can I rigorously prove that $e^{...
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On the charateristic function of random variables
For all random variables that admit a probability density function (PDF), their characteristic function provides an alternative way to completely define its probability distribution.
Why is that? The ...
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Characteristic function for density $f(x) = \dfrac{1}{2(1 + \vert x \vert)^2}, x \in \mathbb{R}$?
Let $X$ be a random variable on $\mathbb{R}$ with the density
$$
f_X(x) = \dfrac{1}{2(1 + \vert x \vert)^2}
$$
I want to find the characteristic function $\varphi_X(t)$. Here is what I've done:
$$...
- 599
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Simple example of using the inversion formula for a non-integrable charateristic function
Background: reading part 2 to this answer, I want to comment asking what the name is for the distribution where $\mu(a,b)+\frac 12\mu(\{a,b\}) = \lim_{n\to +\infty}(2\pi)^{-1}\sum_{j=-n}^{n}\int_0^1\...
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Compute characteristic function
I have to compute the characteristic function of a random variable with density
$$
f(y) = \int_0^{+\infty} \frac{1}{\sqrt{2\pi t}} \exp{\left( -\frac{|x-y|^2}{2t} \right)} p e^{-pt} \, \text{d} t
$$
...
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Convergence of characteristics functions [duplicate]
I have this problem:
If $(a_n)_n$ is a sequence of real numbers and $(e^{i u a_n})_n$ converges to finite limit $g(u)$ for all $u\in I\subset\mathbb{R}$. Show that $(a_n)_n$ converges.
I tried to ...
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How can I use the discrete/fast fourier transform to compute a characteristic function from probability density function?
A characteristic function can be thought of as a fourier transform of a probability density function.
https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)
Using the definition, I ...
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Convergence in probability and differentiability of characteristic function [duplicate]
I am trying to prove the following:
If $(X_1+...+X_n)/n \xrightarrow{n \to \infty} m < \infty$ in probability then the characteristic function $\phi$ is differentiable in zero and $\phi'(0)=im$, ...
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Differentiability in zero of characteristic function and expected value of i.i.i random variables
I would like to prove the following implication:
If the characteristic function $\phi$ is differentiable in zero and $X_1\geq0$ a.s., then $E(X_1)=i\phi'(0)<\infty$.
(This is Exercise 15.4.4 iii) ...
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180
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Prove $a_nX_n+b_n \Rightarrow aX+b$ by means of characteristic functions
I want to solve the following exercise in Probability and Measure, Billingsly [1994]
According to Example 25.8, if $X_n \Rightarrow X$, $a_n \rightarrow a$ and $b_n \rightarrow b$, then $a_nX_n+b_n \...
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$\mathcal X_A · f : \mathbb R^n →\mathbb R$ is $B(\mathbb R^n)-B(\mathbb R)$-measurable and integrable?
Suppose $f : \mathbb R^n → \mathbb R$ is continuous and $A ⊂ \mathbb R^n$ a compact set. Furthermore, $\mathcal X_A$ is
the characteristic function with respect to A, i.e.
$$\mathcal X_A(x) =
\begin{...
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1
answer
45
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Express Random Variable as Sum of Independent Random Variables from Characteristic Function
I am presented with the following Characteristic Function
$$ \phi(\xi) = e^{-2\vert \xi \vert^{1.5} - 0.3 \xi^2 + 1.5i\xi} $$
corresponding to some random variable, and I am tasked with finding the ...
1
vote
0
answers
31
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Confused with Characteristic Function as Fourier Transform of Density Function
The characteristic function of a random variable $X$ is defined as the expectation of the function $e^{itX(\omega)}$ i.e.
$$\int e^{itX(x)}\rho(x)\,dx$$
where $\rho$ is the probability density. How is ...
- 131
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votes
0
answers
109
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Should the characteristic function of a probability distribtion decay to zero?
I am a theoretical quantum physicist trying to find probability distributions for heat transfer in an open quantum system. I am testing an approximate method with an analytically exact one, so I know ...
- 1
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votes
1
answer
37
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$X$ with $\mathbb P(X=1)=\mathbb P(X=-1)=\frac12$ is not infinitely divisible
I want to show that $X$ with $\mathbb P(X=1)=\mathbb P(X=-1)=\frac12$ is not infinitely divisible.
This means I need to show that there $\exists n\in\mathbb N$ such that $\nexists X_1,\dots,X_n$ i.i....
- 616
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0
answers
61
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Uniform continuity of characteristic function of a tight family of measure
I am missing a step in the proof of Theorem 15.22 of Probabiliy Theory by A. Klenke (3rd version).
The theorem states that, given a tight family of probability measure on $\mathbb{R}$, the family of ...
- 525
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votes
1
answer
75
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Use the characteristic function of a geometric function to derive $E[X]=\frac{1}{p}$
Suppose $X$ has geometric distribution $Geo(p)$, I want to use the characteristic function $\phi_{X}(t)=\frac{pe^{it}}{1-(1-p)e^{it}}$ to derive $E[X]=\frac{1}{p}$.
I tried $\phi_{X}(t)=E[e^{itx}]=\...
- 109
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votes
2
answers
180
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Expansion of Characteristic function
I am having troubles in understanding these passages. I am following a Probability course and I took those notes but I did not quite understand the passages.
Be $X$ a random variable, real with $\...
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