All Questions
18,176 questions
-2
votes
0
answers
7
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
43
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
2
votes
0
answers
32
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 \...
- 400
0
votes
0
answers
16
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
35
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
66
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
25
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
104
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
64
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
38
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
273
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
1
vote
0
answers
90
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 ...
- 400
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
55
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
90
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
82
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
77
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
224
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
$$X_0=0\quad \mbox{ and } \quad\...
- 1,389
0
votes
1
answer
82
views
Median of cardinality of set union
Let $U$ be an arbitrary finite universe (you can just think of it as $[N]=\{1,2,\ldots,N\}$), and $\mathbf{S} = (S_i)_{i \in [n]}$ ($S_i \subseteq U$) be the sets that we are drawing from. Define a ...
- 29
3
votes
1
answer
84
views
What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?
Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having ...
- 73
12
votes
1
answer
394
views
Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\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
157
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
292
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
85
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
37
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
62
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}{\...
- 835
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 ...
- 400
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 (\...
- 6,155
5
votes
1
answer
375
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\...
- 4,135
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 ...
- 511
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
79
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
1
vote
0
answers
48
views
What makes the generalized projection different than metric on a Banach space?
I have came across the notion of generalized projection in Banach spaces, introduced by Ya. Alber and has seen many iterative algorithms being solved by using this projection. It helps in finding the ...
- 85
0
votes
0
answers
53
views
Spectral theory of compact operator for quasi-Banach spaces
Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
- 938
4
votes
0
answers
116
views
Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
- 41