All Questions
Tagged with cv.complex-variables pr.probability
65 questions
3
votes
1
answer
116
views
Interpretations of analytic continuations of CDFs to complex probabilities
Are there notable cases where analytic continuations of cumulative distribution functions to complex arguments have a meaningful interpretation or are otherwise useful?
If a one dimensional CDF is ...
- 7,525
2
votes
0
answers
157
views
Conformally mapping between the upper half complex plane, and the plane with a tree on spatial points removed
A stochastic process such as SLE$_{\kappa}$ can be defined by taking the scaling limit of a curve in the upper half complex plane: put simply, one removes a line segment, then another, $n$ times, each ...
- 640
5
votes
2
answers
791
views
How to calculate an integral over the complex unit sphere
We want to calculate the following integral over the complex unit sphere $S^{2n-1}$:
$$\int_{S^{2n-1}} \frac{1 }{|1 - \langle z, \zeta \rangle|^2} \, d\sigma(\zeta),$$
where $ z $ is a fixed point in ...
- 113
0
votes
0
answers
59
views
Convergence of Liouville correlation functions
A key object in Liouville conformal field theory is the random Liouville measure $M$ defined heuristically as $M(d^2x) = :e^{2bX(x)}: d^2x$, where $X$ is a Gaussian free field and $:e^{2bX}:$ denotes ...
user528837
0
votes
0
answers
38
views
Reference on multifractal complex measures?
This is a cross-post of this physicsSE post; I am also posting it here since this question lies at the boundary of both physics and math.
I am learning about multifractal formalism recently. It seems ...
- 715
0
votes
1
answer
103
views
Probabilistic bounds of random polynomials
This is follow-up question to my previous question about the expected number of roots .
I am considering a random polynomial given by $$p(z) = \sum_{i=0}^{n} a_i z^i$$,
where each coefficient } $a_i$ ...
- 826
2
votes
1
answer
133
views
Expected fraction of roots in the unit disc of random polynomial with Gaussian coefficients
I am trying to find the expected fraction of roots located in the unit disc for a random polynomial with Gaussian coefficients. Given a random polynomial
$$P(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^...
- 826
4
votes
1
answer
205
views
Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$
Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...
- 730
1
vote
2
answers
354
views
Reference request and clarification for Central Limit Theorem for complex random variables
I'm looking for a reference and a proof of the following version (or eventually a more general version) of the Central Limit Theorem for complex random variables.
Theorem. Let $Z_1, Z_2, \dots, Z_n$ ...
- 361
10
votes
1
answer
1k
views
Proving the Replica Trick works
The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit
$$
\log(Z) = \lim_{n\to 0}\...
- 385
0
votes
0
answers
79
views
Geometry of inner products between the unit vector and several given vectors
Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e.,
$$
\mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...
- 550
6
votes
3
answers
855
views
Series involving power of the index
How to prove the following identity
$$ \sum_{n=1}^{\infty} \frac{n^{n-1} e^{-n}}{n!} = 1$$
analytically (which can be confirmed with $Mathematica$)? The standard trick for geometrical series does not ...
- 77
0
votes
1
answer
443
views
Expectation of complex random variable
I am researching frequency offset estimation and ended up reading a paper "Cramer-Rao Lower Bound on Frequency Offset Estimation Error in OFDM Systems With Timing Error Feedback Compensation"...
3
votes
2
answers
427
views
Request for recommendation in probability and complex analysis
Could somebody kindly recommend to me some books which deal with the applications of the probabilistic method to problems in real and complex analysis or which consider probabilistic versions of some ...
- 826
4
votes
0
answers
179
views
As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?
As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
0
votes
1
answer
255
views
Sufficient conditions for decomposition of a bounded random variable into several small pieces
Given a random variable $X$ with $\mathsf{supp}\, X \subseteq [0,1]$ and $n$ positive numbers $h_1,\cdots,h_n$ with $\sum_{i=1}^n h_i=1$, I want to know some sufficient conditions for decomposing $X$ ...
- 550
3
votes
1
answer
730
views
Conditional independence in measure-theoretic terms
Let $\Omega$ be a compact Hausdorff space in $\mathbb{C}^n$. Let $\sigma_\Omega$ be the Borel sigma algebra on $\Omega$. Let $\zeta: \Omega\longrightarrow\partial \mathbb{D}$ be a non constant ...
- 149
1
vote
0
answers
107
views
Comparison of two Fourier transforms
I am looking for $\delta>0$, such that
$$
\delta \int_{-\infty}^{\infty} \exp(its)
{ \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\
\int_{-\infty}^{\infty} \exp(its)
{ \Gamma (it+1)\over \...
- 93
6
votes
1
answer
239
views
Positivity of $ \int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it} { \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt$
I have the following function
$$
\int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it}
{ \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt
$$
where $1<\beta<2$, $v>0$. Need ...
- 93
4
votes
0
answers
136
views
Independent inner functions on the unit disk
This problem cropped up in a paper that I am writing and I have thought about it for too long to no avail: let $\mathbb{D}$ be the open unit disk in the complex plane and suppose $\varphi:\mathbb{D}\...
- 41
0
votes
0
answers
94
views
Theorem 5.3 ([Okounkov-01]) in Borodin and Gorin's lecture note
In this lecture note: https://arxiv.org/pdf/1212.3351.pdf, Theorem 5.3(P28):
Suppose that the $\lambda \in \mathbb{Y}$ is distributed according to the Schur measure $\mathbb{S}_{\rho_1; \rho_2}$. ...
- 288
3
votes
1
answer
219
views
Reference request: The transform of a bounded random variable has a zero in the complex plane
Together with coauthors I'm working on a paper where we use the following Proposition:
If a real-valued random variable $X$ has bounded support, then except in the trivial case that $X$ has all ...
- 5,492
6
votes
1
answer
128
views
The distribution of the area of a region cut out by chordal SLE?
Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$.
For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \...
- 1,462
5
votes
1
answer
258
views
Find a function $F$ on $[0,1]$ with moments decaying as $(\ln n)^{-n}$
Let $F:[0,1]\to\mathbb{R}$ be a measurable function such that
$$
\mu_n(F)=\int_0^1F(t)t^ndt\sim\frac1{(\ln n)^n}\quad\mbox{as}\quad n\to\infty.
$$
More precisely,
$$
0<c<|\mu_n(F)|(\ln n)^n<...
- 1,959
4
votes
3
answers
654
views
Non combinatorial random matrix theory
I am learning random matrix theory. Unfortunately I do not like combinatorics, and have never really been good at it. But I found that random matrix theory has heavily relied on combinatorics, ...
- 267
2
votes
0
answers
171
views
Distribution of random hyperplanes in projective spaces
Let $X\subset \mathbb{CP}^{N-1}$ be a smooth subvariety of dimension $n$. Assume that $X$ is not contained in a hyperplane of $\mathbb{CP}^{N-1}$.
Let $\mu$ be a smooth probability measure on $X$. ...
- 404
2
votes
1
answer
84
views
Is the distribution of the real part of product of two independent complex variates exponential?
Trying to find the pdf of the real part x of the product $z_1z_2$ of two uncorrelated complex random Gaussian variates . The pdf of the modulus $r \equiv |z_1z_2|$ is known
$ f_r(r)=rK_0(r)$ from ...
- 51
3
votes
1
answer
206
views
Capacity of a unit disk with a small bump
Let $A_r = \{z\in\mathbb{C}: |z|\leq 1\}\cup\{z\in\mathbb{C}: |z-1|\leq r\}$ be the unit disk with a small "bump" (I'm interested in the regime $r\to 0$). What can be said about the logarithmic ...
- 1,897
1
vote
0
answers
101
views
Density of Geometric Stable distribution
If we define
$$
\psi(t|\alpha, \beta, \gamma, \mu) = -it\mu+|\gamma t|^\alpha(1-i\beta \mathrm{sgn}(t) \Phi)
$$
with
$$\Phi = \tan \frac{\pi \alpha}{2} \mathbf{1}_{\{ \alpha \neq 1 \} } - \frac{2}{\pi}...
1
vote
1
answer
107
views
Bounding function by random sampling
Let $f(x): \mathbf{C} \to \mathbf{C}$ be a complex valued polynomial of degree $n$, and suppose that for a random point $z$ sampled uniformly from the unit disk $|z|\leq 1$ we have
$$\mathbf{P}_{z}( |...
- 445
14
votes
1
answer
3k
views
How is the "conformal prediction" conformal?
The question is clarified by Prof.V.Vovk. See his answer below for discussion.
Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
- 8,071
2
votes
0
answers
103
views
measures in infinite dimension space of entire functions [closed]
It is known that there is no canonical generalization of Lebesgue measure in infinite dimension of function spaces. Since it seems that the space of (transcendental) entire function seems improtant ...
- 1,706
1
vote
1
answer
148
views
Intuitional feeling of harmonic measure on one-third Cantor set
It is known that the harmonic measure on classical one-third Cantor set has Hausdorff dimension strictly less than $\frac{\log 2}{\log 3}$. Even harmonic measure has a close relation with brownian ...
- 1,706
1
vote
1
answer
3k
views
How to calculate the expected value of complex-valued random variable? [closed]
Suppose $\theta_1,\theta_2,\cdots, \theta_n$ are independent and identically distributed (i.i.d.) real-valued random variables and here we specifically consider the uniform distribution in the ...
- 19
0
votes
0
answers
260
views
Concluding that the Poisson kernel is indeed the Cauchy distribution?
See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
- 13
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
6
votes
1
answer
750
views
Roots of characteristic function of "reciprocal gamma measure"
Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...
- 133
3
votes
1
answer
902
views
Is the integral always nonzero?
Let
$$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$
where
$$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < n,\qquad\qquad\qquad\qquad\...
- 128k
3
votes
1
answer
216
views
Determining the Fourier transform
Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a ...
- 277
6
votes
1
answer
536
views
Bound on the sum of arguments
Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...
- 128k
1
vote
2
answers
689
views
"Convolution" for Multiplying Random Variables
The following situation arises frequently in probability.
Suppose we have two independent continuous random variables $X$ and $Y$ and we consider their sum, $Z=X+Y$. Then the pdf of $Z$ is the ...
- 3,979
7
votes
5
answers
682
views
Bound on sum of complex summands involving binomial coefficients
I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have $|x|...
- 93
5
votes
1
answer
429
views
Why are the angular differences of these random complex polynomial coefficients almost constant?
This is based on მამუკა ჯიბლაძე's (not-)answer here. I guess it is better to make up a new thread for it.
Let me repeat the setup here: We consider polynomials whose complex roots are randomly ...
- 13.4k
5
votes
0
answers
275
views
Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?
Recent questions showed that roots of a random polynomial tend to lie on the
unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
- 151k
14
votes
4
answers
3k
views
A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst
I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability.
A quick google search gave a lot of references on SLE ...
- 2,154
430
votes
16
answers
65k
views
Why do roots of polynomials tend to have absolute value close to 1?
While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
- 48.8k
2
votes
0
answers
221
views
why is this result about Gaussian analytic functions equivalent to the Crofton formula
I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function.
Definition A Gaussian analytic function $...
- 22.8k
4
votes
1
answer
207
views
M-Wright function asymptotics
Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, $0<\nu<...
- 424
0
votes
0
answers
134
views
Mittag-Leffler function and Laplace Integral
Let $E_{\alpha}(z)\triangleq \sum_{n=0}^{\infty} \frac{z^n}{\Gamma(\alpha n + 1)}$ be the Mittag-Leffler function.
I am looking for a full proof of the following fact (a reference to a proof in the ...
- 424
2
votes
1
answer
850
views
Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)
Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is
$$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z \right|...