All Questions
18,179 questions
0
votes
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5
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Hypergeometric functions in Wireless communication: Seeking guidance for Performance analysis
I am transitioning from pure mathematics to wireless communications and am particularly intrigued by the mathematical challenges in analyzing Nakagami-m fading channels. These channels are widely used ...
- 41
-4
votes
0
answers
48
views
Is the probability of the Euler-Mascheroni constant being rational large? [closed]
Let $\gamma$ denote the Euler Mascheroni constant. Show there exist infinitly many polynomials $p\in \mathbb{Z}[x]$ such that $p(x) = \gamma$. Further, show one can calculate a probability that some ...
6
votes
0
answers
106
views
Size doubling amoeba
Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$.
A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p &...
- 6,215
-3
votes
0
answers
23
views
Convergence of measures in the Lévy–Prokhorov metric and weak convergence of measures
How to prove that over R the convergence of measures in the Levi-Prokhorov metric is equivalent to the weak convergence of measures
- 1
1
vote
0
answers
85
views
Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
- 405
3
votes
0
answers
76
views
Order of $\mathbb{E}[ \max_i |x_i + z_i| - \max_i |z_i|]$
Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$.
Define
$$
F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big]
$$
If $\|x\|_\infty \...
- 420
0
votes
0
answers
18
views
A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
- 51
1
vote
0
answers
45
views
Square-integral involving Brownian bridge
Let $B(t)$ be a standard Brownian bridge on $[0,1]$. Let $x>0$ be a (small) parameter. What is the distribution of
$$
\int_0^{1-x} \left( B(t + x) - B(t) \right)^2 dt?
$$
As noted I am interested ...
- 2,207
-3
votes
0
answers
70
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
-1
votes
0
answers
26
views
Estimate the value of the PDF $P(f)$ at the minimal $f_0$ of the random-variable function $f(\mathbf{x})$
Let $f(\mathbf{x})=f(x_1,x_2,\dotsc,x_N)$ with $N>2$ be a real and continuous function and $f(\mathbf{x})\ge f_0$ for any $\mathbf{x}\in\mathbb{R}^N$. Now let $x_1,x_2,\dotsc,x_N$ be the i.i.d. ...
- 375
2
votes
0
answers
109
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
0
votes
0
answers
66
views
$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable
Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported.
I am wondering if we can ...
- 91
1
vote
0
answers
39
views
Can conditional distributions with respect to a sufficient sub-$\sigma$-algebra be represented by a single Markov kernel?
Let $(\Omega, \mathcal{F})$ be a measurable space, and let $\mathcal{P}$ be a collection of probability measures on this space. A sub-$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is said to be ...
- 111
3
votes
1
answer
139
views
Surjectivity of pushforward on image
Let $\mathcal X\subseteq\mathbb R^m$ be a Borel measurable set. $\Phi:\mathcal X\to\mathbb R^n$ be a continuous mapping and $\mathcal Y = \Phi(\mathcal X)\subseteq\mathbb R^n$ its image. Let $\mathcal ...
- 345
4
votes
1
answer
277
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
2
votes
0
answers
97
views
+100
Inequalities for norm of centered Gaussian and uncentered Gaussian
Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm.
Let $x \in \mathbb{R}^n$ and define
$$
F(x) = \mathbb{E}[\|x + g\| - \|g\|].
$$
I am wondering if it is possible to ...
- 420
1
vote
1
answer
54
views
Proving bound on expectation of likelihood ratio involving mixtures
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
- 13
1
vote
0
answers
56
views
Quantitative multivariate CLT from quantitative CLT of linear combinations
Suppose $Z_1, \ldots, Z_k$ are random variables with mean $0$ and variance $1$ that are "approximately jointly Gaussian" in the sense that for any scalars $c_1, \ldots, c_k$, we have that $\...
- 111
3
votes
0
answers
91
views
About BMO space on smooth open bounded domain
Let $\Omega$ be any open domain in $\Bbb R^d$.
Define the $\text{BMO}(\Omega)$ space as
$$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\},
$$
...
- 1,101
3
votes
0
answers
83
views
+50
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 73
2
votes
0
answers
79
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
2
votes
0
answers
227
views
A deceptively simple regularity problem for functions on the plane
By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer:
Consider a twice ...
- 989
0
votes
0
answers
24
views
Characterisation of a family of continuous martingales
I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that
$$X_0=0\quad \mbox{ and } \quad\...
- 1,399
0
votes
1
answer
82
views
Median of cardinality of set union
Let $U$ be an arbitrary finite universe (you can just think of it as $[N]=\{1,2,\ldots,N\}$), and $\mathbf{S} = (S_i)_{i \in [n]}$ ($S_i \subseteq U$) be the sets that we are drawing from. Define a ...
- 29
3
votes
1
answer
85
views
What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?
Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having ...
- 73
12
votes
1
answer
394
views
Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\int_S ...
- 24.2k
3
votes
0
answers
67
views
Effective action of unbounded operators on subspaces outside their domains of definition
Consider a densely defined, self-adjoint operator
$$
H: \mathcal{D} \rightarrow \mathscr{H}.
$$
Assume for simplicity that $H$ is nonnegative.
We want to effectively restrict this operator $H$ to a ...
- 133
0
votes
1
answer
159
views
Weak convergence of $f(x,e^{itx})$
This is the desired result (what I want to prove):
$$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$
Given that $f\in C([a,b]\times\{e^{i\...
- 233
9
votes
1
answer
297
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
0
votes
0
answers
46
views
Amenability of locally convex algebras
Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.).
Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
- 2,605
1
vote
0
answers
86
views
Density of a subset of Schwartz space in the fractional Sobolev space
It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...
0
votes
0
answers
38
views
Bounding the error of a truncated moment problem
Let $\{x_{i}\}_{i=1}^{\infty}$ be a non-increasing sequence of non-negative real numbers, and let $\{y_{j}\}_{j=1}^{B}$ be a non-increasing sequence of non-negative real numbers, where $B$ is a finite ...
- 433
1
vote
1
answer
151
views
Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
- 113
2
votes
0
answers
63
views
Bessel spaces and Triebel Lizorkin
It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
1
vote
0
answers
58
views
duality of sobolev spaces. Representation of elements in the dual
I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
0
votes
0
answers
42
views
questions on stochastic kernels and pushforward operator
Let $f:X \rightarrow \Delta (Y)$ and $g:X \rightarrow \Delta (X)$ be two kernels. For any bounded measurable function $h_Y:Y \rightarrow \mathbb{R},$ define $F(h_Y):X \rightarrow \mathbb{R}$ such that ...
- 1
-1
votes
0
answers
35
views
Different definition of Feller semi-group
(This is a crosspost of a question on MathStackExchange which did not receive any answer.)
Let $E$ be a locally compact metric space, let $C_0(E)$ be the set of real-valued continuous functions of $E$ ...
-1
votes
1
answer
93
views
Variance of bins for N balls into M bins [closed]
If I throw N balls independently into M bins with uniform probability, the expected mean of the M bins is N/M balls.
What is the expected variance of the M bins?
I was thinking of what bin size I ...
4
votes
0
answers
90
views
Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 825
2
votes
1
answer
147
views
Lower bound in the singularity of random Bernoulli matrices
Let $A_n$ be a random $n \times n$ matrix with entries in $\{-1, +1\}$. As usual, "random" here means with respect to the uniform measure over such matrices.
The strong version of the ...
- 420
0
votes
0
answers
90
views
How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?
I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak*
-Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
- 113
3
votes
0
answers
80
views
Asymptotics of number of running maxima of iid random variables
Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities.
Given a realisation $\omega$ of the random variables, we say that $X_i (\...
- 6,215
5
votes
1
answer
376
views
Convergence of random functions
Suppose I have a sequence of random continuous functions, $f^{n} : [0, t] \to \mathbb{R}$. Suppose there also exists a random continuous function, $f: [0, t] \to \mathbb{R}$, defined on the same ...
- 85
0
votes
1
answer
124
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,143
1
vote
0
answers
146
views
integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 521
0
votes
0
answers
31
views
Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales
Does anybody know a reference for the following theorem?
Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale.
Then, for any constant $c > 0$, the event $(\exists
> t)\, X_t \...
- 183
2
votes
0
answers
63
views
Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$
In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
0
votes
1
answer
239
views
Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]
Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
4
votes
0
answers
101
views
There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions
Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$.
Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
- 3,669
1
vote
0
answers
81
views
Markov Chain that maximises the entropy creation rate
I am working on MERW (Maximal entropy random walk) for a project.
I want to show that given a graph G, there is $\textbf{only one}$ aperiodic markov chain on G that maximises the entropy creation rate ...
- 11