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Name of PDF? - projecting uniform probability distribution on the unit circle to the x-axis

Consider a uniform probability distribution on a circle of radius r, i.e. $\{(x,y) \in \mathbb{R}^2: x^2 + y^2 = r^2 \}$.If we wish to project onto the x-axis, we can consider each point on the circle ...
SSD's user avatar
  • 225
4 votes
0 answers
168 views

Sum of Independent Laplacian Variables

I have $N$ Independent Random variables Laplacian distributions with $\mu=0$ and positive $b=\sigma^2/2$. I also have dominant random variable $(X_s)$ with Laplacian distribution with $\mu=0$ and $b=\...
lone_wolf's user avatar
  • 166
1 vote
0 answers
63 views

Sampling from a distribution characterized by its characteristic function

Consider the following measure: $$d\nu (x)=\mathbb 1_{(0,1)} (x) \frac 1 {x^{2}}$$ Now, define $X$ with characteristic function given by: $$\varphi_{X}(t)= \exp\left\{ \int_{\mathbb R} [e^{itx}-1 - ...
PSE's user avatar
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0 answers
75 views

A problem of weak convergence of stochastic processes

Let $(X_1,X_2, ....)$ be a infinite sequence of random variables. Supoose that te sequence is strictly stationary. For all $n$, define the following infinite array: $$ \begin{bmatrix} Y_1 & Y_2 &...
Fam's user avatar
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0 answers
22 views

An MA model has MA characteristic polynomial $(1 − 1.4x + 0.3x^2 )(1 + 0.5x^{12})$, obtain the model [duplicate]

When the characteristic polynomial of a moving average s model is given as $(1 − 1.4x + 0.3x^2 )(1 + 0.5x^{12})$, how to obtain the MA model and then calculate the ACF of this model? I am expecting ...
Kavindu Ravishka's user avatar
2 votes
3 answers
164 views

If $X$ and $Y$ are independent random variables with $X+Y\stackrel{d}{=}X$, then show that $\mathbb P(Y=0)=1$ [closed]

Show that if $X$ and $Y$ are independent random variables with $X+Y\stackrel{d}{=}X$, then show that $\mathbb P(Y=0)=1$. Can the independence condition be dropped? I could solve the first part using ...
Sayan Dutta's user avatar
0 votes
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Name of a property of characteristic functions (Fourier transforms)

I'm a scientist but not a professional mathematician and in this question, I asked about a possible typographical error in an article on round-off error that I've been reading in the journal ...
CrimsonDark's user avatar
0 votes
1 answer
81 views

Inverting a characteristic function if the integral of the modulus of the cf is infinity

I'm reading a lecture slide that starts by asking if there's a way to invert a characteristic function $\psi_X$ if $\int|\psi_X(t)|~\mathrm{d}t = \infty$. From my reading, the slide then provides a ...
johnsmith's user avatar
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5 votes
2 answers
1k views

Find mean and variance using characteristic function

Consider a random variable with characteristic function $$ \phi(t)=\frac{3\sin(t)}{t^3}-\frac{3\cos(t)}{t^2}, \ \text{when} \ t \neq0 $$ How can I compute the $E(X)$ and $Var(X)$ by using this ...
Alex He's user avatar
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Two non-obviously identical random variables that can be shown to be identical via their characteristic functions

It is well known that a characteristic function (CF) is uniquely associated with a probability density function (PDF). Knowing this, I was nonetheless intrigued by a remark in a video I have been ...
CrimsonDark's user avatar
3 votes
2 answers
486 views

Show the Binomial distribution approaches a Normal distribution (using characteristic function)

Let $X_n = Bin(n,p)$. We fix $p$, and we want to show that as $n \to \infty$, $\frac{X_n-np}{\sqrt{np(1-p)}}$ converges to $N(0,1)$ in distribution. And I want to show this using characteristic ...
Phil's user avatar
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1 answer
255 views

Density from characteristic function: Durrett example 3.3.8 and 3.3.9

Letting $\varphi(t)$ be the characteristic function for the probability measure $\mu$, we know if $\int \left|\varphi(t)\right|dt < \infty$, then $\mu$ has density function $$f(y) = \frac{1}{2\pi} \...
Phil's user avatar
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1 answer
48 views

Investigating the presence of unit root in the following $X_t$

I am given a model and need to calculate the unit-root of $X_t$ but it seems that there is no unit-root. The model is given: $X_t = (x_{1t},x_{2t})'$ $$\Delta X_t = \alpha \beta ' X_{t-1} + \epsilon_t ...
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1 vote
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121 views

Find the characteristic function of a random variable involving a compoud poisson distribution

Let $\{\xi_i \}_{i=1}^{k}$ be a finite sequence of independent r.v. such that $\xi_{i} \sim F_i$. For each $i$, let: $\{\xi_{i,j}\}_{j=1}^{\infty}$ be a sequence of copies of $\xi_i$, that is: $\xi_{...
Fam's user avatar
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1 vote
1 answer
34 views

Characteristic function of linearly transformed random variables with extracted factor

Given is a sequence of independent random variables $X_1, X_2,\ldots, X_n$ with characteristic function $\varphi_{X_i}(t)$. The characteristic function of $Y = \sum_{i=1}^n a_i X_i$, where the $a_i$ ...
granular_bastard's user avatar
2 votes
0 answers
115 views

How to calculate the covariance matrix in (Xu and Knight 2010)?

Setup I'm reading (Xu and Knight 2010), which is a paper about estimating finite Gaussian mixture models using the CECF (Continuous Empirical Characteristic Function) method. The basic idea is to ...
ForceBru's user avatar
  • 330
2 votes
0 answers
138 views

Stationarity of AR(2) model

I have one question, which is related to a solution presented in "A proof for the stationarity of an AR(2)" by Christoph Hanck. Since I still cannot write comments (I need at least 50 points)...
Igalala's user avatar
  • 119
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1 answer
171 views

Derivation of moment generating function for limiting distribution of sum of logbeta distributed variables

A sum of logbeta distributed variables occurs in this question Distribution with a given moment generating function Let, $X_j \sim Beta(j\sigma, 1-\sigma)$, $Y_j = -\log(X_j)$ and $S_n = \sum_{j=1}^n ...
Sextus Empiricus's user avatar
2 votes
1 answer
94 views

Finding $\mathbb{E}|X|$ for variance-gamma random variable

Assume $X \sim f_{X}(n,\mu, \sigma)$ is a Variance-Gamma random variable. The density function involves a modified Bessel function, therefore is not that trivial to handle. I'm looking for $\mathbb{E}|...
runr's user avatar
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0 answers
182 views

characteristic function of standard lognormal distribution

What is the characteristic function of standard lognormal distribution? By "$X$ yields the standard lognormal distribution", I mean $\log(X)\sim N(0,1)$. I understand that the moments of $X$ ...
Tan's user avatar
  • 1,499
2 votes
1 answer
447 views

Characteristic Function and Quantile Function?

Characteristic functions (cf) are closely related to cdfs and pdfs of random variables, for example cf is the Fourier transform of the pdf Inversion formulae from Lévy and Gil-Pelaez Question: Is ...
Alex's user avatar
  • 347
1 vote
1 answer
82 views

tails from cumulant generating function

I am trying to obtain finite-sample tail estimates using the mgf. I found a snippet in a google books preview of a formula giving the survivor function by some kind of inversion of the cumulant ...
Hasse1987's user avatar
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0 answers
29 views

$lim_{n→∞}P((/1√n)\sum\limits_{i=1}^n(−1)^{i}X_{i}\leq$x) equals?

Let X1, X2, . . . be a sequence of i.i.d. random variables with variance 2. Then for all x, $lim_{n→∞}P((/1√n)\sum\limits_{i=1}^n(−1)^{i}X_{i}\leq$x) equals $(A) Φ(x√2)$ $(B) Φ(x/√2)$ $(C) Φ(x)$ $(D) ...
Muskaan Madan's user avatar
3 votes
1 answer
3k views

Characteristic function and Fourier transform for a discrete random variable!

Let $\phi_{x}(t)= E [ e^{itx}]$ be the characteristic function If X is a continuous random variable, then: $\phi_{x}(t)= E [ e^{itx}] = \int e^{itx} f(x)dx$ (being $f(x)$ the probability density ...
Ilya_Curie's user avatar
0 votes
0 answers
75 views

Goodness of fit using characteristic function

When assessing goodness of fit of a model, one can use, for example, a Q-Q plot of empirical vs theoretical distributions. But how does one perform a GoF assessment when there is no closed form ...
Confounded's user avatar
1 vote
0 answers
54 views

Why do we use exponent in characteristic function? [closed]

A student who is attending probability 101, learned about normal distribution and generating functions recently. We are given a "generating function" as follows: $$G(t)=\langle e^{itx}\...
hrmck's user avatar
  • 11
4 votes
1 answer
663 views

What is the characteristic function of a rectified Normal distribution?

Rectified Normal distribution is a hybrid distribution with the following pdf: $f(x;\mu ,\sigma ^{2})=\Phi (-{\frac {\mu }{\sigma }})\delta (x)+{\frac {1}{{\sqrt {2\pi \sigma ^{2}}}}}\;e^{{-{\frac ...
Ramin Barati's user avatar
0 votes
1 answer
56 views

How to perform van Zyl's (2018) ecf-based normality test in R [closed]

I have been reading quite a bit on empirical characteristic function (ecf) based normality tests, but cannot find any functions to perform such a test in R and lack the mathematical ability to figure ...
Conrad's user avatar
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1 vote
0 answers
221 views

How to find the characteristic function of a function related with Shannon entropy?

A random variable X is distributed with a known probability distribution $p(x)$. Suppose that $x$ is sampled in an independent and identically distributed process and with the results $\vec{x}=(x_1,...
user557651's user avatar
4 votes
2 answers
417 views

Expected of number of discrete uniform variables whose sum is bigger than k (from characteristic function of discrete Irwin–Hall distribution?)

The problem Imagine we keep uniformly drawing $n$ integers $X_i$ from {0, 1, ..., 9} so that their sum is more than $10$. For instance, one draw would be {1, 0, 2, 5, 3}, hence $n=5$, and repeat this ...
psyguy's user avatar
  • 311
1 vote
0 answers
43 views

How can we write the below characteristic function?

Let us assume that $X$ is a random variable and $a$ is a constant. Now suppose $Y=a+bX$, what would the characteristic function of $Y$ would be? Is it? \begin{eqnarray} \mathbb{E}_X\left[\exp(iuY)\...
Carl's user avatar
  • 1,226
2 votes
0 answers
120 views

On likelihood functions and characteristic functions

Let me preface this by saying that if someone manages to provide a solution to my problem, I will forever be indebted to them, as this problem has driven me crazy. Let us first assume that the ...
Carl's user avatar
  • 1,226
4 votes
1 answer
175 views

A question about characteristic functions

A probability 101 question. We know that if two variables $X$ and $Y$ are independent then the characteristic function $\phi_{X+Y}(u)$ can be written as \begin{equation} \phi_{X+Y}(u)=\phi_{X}(u)\phi_{...
Carl's user avatar
  • 1,226
1 vote
0 answers
74 views

A simple question about expectations

@psboonstra This is a valid point. After posting it, I too realized that the question is ill-posed. I attempted to oversimplify a problem that I had encountered in finding a characteristic function of ...
Carl's user avatar
  • 1,226
1 vote
0 answers
91 views

Question about the log-normal distribution

The main object of my question is this: if $X$ has a log-normal distribution, $Y = X + Z$ and $Y$ has the same distribution as that of $Z^2$ (in other words, $F_{Z^2} = F_{X+Z}$) and $X, Z$ are ...
Vincent Granville's user avatar
13 votes
1 answer
487 views

Random variables $X, Z$ such that $Z$ and $\sqrt{X + Z}$ have the same distribution?

I am looking for the distribution of a random variable $Z$ defined as $$Z = \sqrt{X_1+\sqrt{X_2+\sqrt{X_3+\cdots}}} .$$ Here the $X_k$'s are i.i.d. and have same distribution as $X$. 1. Update I ...
Vincent Granville's user avatar
1 vote
0 answers
103 views

Difference between characteristic function and F-transform

I'm struggling to understand the difference between this two functions. I have this condition: $P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\...
Marco Pittella's user avatar
1 vote
1 answer
89 views

Finding the characteristic function of a simple linear PDF

X is a random variable with a pdf of $ f(x) = \begin{cases} x/2, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases}$ I tried finding the characteristic function of this but ended up with ...
Maharero's user avatar
1 vote
0 answers
425 views

How to pass from {Probability density function, convolution} to {Probability density function, characteristic function}?

In Forsman, W.C. (1986) "Polymers in solution: theoretical considerations and newer methods of characterization", Springer, New York. https://www.springer.com/la/book/9780306421464 page 24, it states:...
Erdogan CEVHER's user avatar
1 vote
2 answers
3k views

How to modify the mean and variance/dispersion of a given distribution

I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ...
sets's user avatar
  • 317
2 votes
2 answers
129 views

Expected value of $e^{sP}$ where s is a complex number and $P$ is a Poisson rv

For each positive integer $N$, let $ B_N$ be a binomial $(N,1/3)$ random variable and $P$ be a Poisson(5) random variable. I am trying to understand the statistics of $B_P$. Could someone please hint ...
pickle_lover's user avatar
0 votes
0 answers
21 views

Proving the characteristic function involving sequence of 2 iid variables [duplicate]

We have an iid sequence with: $X_1,X_2,...$ with mean $\mu$ and variance $\sigma^2$. We have another similar sequence: $$Y_1,Y_2,...$$. We have a sequence $U1,U_2,...$ where, $U_n=1/\sqrt(2(n\sigma^...
StatisticsPersonInTraining's user avatar
5 votes
4 answers
592 views

Proof of Convergence in Distribution with unbounded moment

I posted the question here, but no one has provided an answer, so I am hoping I could get an answer here. Thanks very much! Prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{...
NamelessGods's user avatar
9 votes
2 answers
3k views

When to prefer the moment generating function to the characteristic function?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X : \Omega \to \mathbb{R}^n$ be a random vector. Let $P_X = X_* P$ be the distribution of $X$, a Borel measure on $\mathbb{R}^n$. The ...
Artem Mavrin's user avatar
  • 4,067
3 votes
0 answers
33 views

finding process corresponding to laplace transform

I have a positive stochastic process $X(t)$ with Laplace transforms $$ \mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b} $$ One can clearly see that the ...
lbf_1994's user avatar
  • 528
3 votes
1 answer
77 views

Good book on characteristic functions that includes the CF-proof of the CLT

The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ...
6 votes
2 answers
1k views

Sampling from characteristic/moment generating function

Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ...
lbf_1994's user avatar
  • 528
6 votes
2 answers
159 views

Characteristic function inequality

Random variable $X$ and its characteristic function $\phi_X(t)$ then $$\Pr\left(|X|>\frac2T\right) \leq 2\left(1 - \frac1{2T}\int_{-T}^{T}\phi_X(t)dt\right) $$ I cannot find a way how to ...
Ethan's user avatar
  • 483
5 votes
1 answer
260 views

Noncentral chi² with a noncentral chi² noncentrality parameter

Denote by $\chi^2(\nu,\lambda)$ the noncentral chi-square distribution with degrees of freedom $\nu$ and noncentrality parameter $\lambda$. If $\Lambda \sim \chi^2(\nu, 2\theta)$ and $(X \mid \Lambda)...
Stéphane Laurent's user avatar
4 votes
1 answer
415 views

Application of Central Limit Theorem - Uniform Distribution

The following question I found on an old exam: Given $n$ i.i.d. random variables $X_k$, $1 \leq k \leq n$, with uniform distribution on $[-1,1]$, it is easy to compute the characteristic function of ...
Mitch Baker's user avatar