All Questions
18,178 questions
0
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503
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When are operators extended by linearity bounded?
Greetings.
Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite
dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
- 129
0
votes
0
answers
458
views
Bounding mutual information given bounds on pointwise mutual information
Suppose I have two sets $X$ and $Y$ and a joint probability distribution over these sets $p(x,y)$. Let $p(x)$ and $p(y)$ denote the marginal distributions over $X$ and $Y$ respectively.
The mutual ...
- 1
0
votes
0
answers
163
views
"Reverse" stochastic dominance
Let $\mu$ and $\mu'$ be probability measures on $\lbrace0,1\rbrace^\Lambda,\:\: \Lambda:= {\lbrace 0,1,\ldots,n\rbrace}$. Assume that
$\mu(X_i=1|X = \zeta \text{ on } \Lambda \setminus \lbrace i\...
- 1,491
0
votes
0
answers
301
views
Lifting of product of a Banach algebra
Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...
- 1,222
0
votes
2
answers
383
views
"X \in \cdot" in Probability Measure [closed]
My question is quite simple, but I was unable to find an answer by googling, since you can't exactly google syntax. What does the $\in \cdot$ mean in:
$$\lim_{n\to\inf}||P(S_n\in\cdot)-P(S_n+k\in\cdot)...
- 3
0
votes
0
answers
293
views
Open Jackson network with deterministic arrivals.
Dear Friends,
Is there any known Jackson-like theorem for an open Jackson network with deterministic arrivals?
Thanks,
Michael.
- 85
0
votes
0
answers
319
views
Estimating a multinomial sum
I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
\...
0
votes
1
answer
578
views
One-Variable Optimization Problem
$W_{opt}=\arg \{\max(\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha )\}$
subject to $\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$
We should find analytically the optimal $...
- 23
0
votes
1
answer
1k
views
Linear Mapping and integration
I have been reading the paper - "Introduction to Quantum Fisher Information".
In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows:
Let $D \in M_n$ be a ...
- 145
0
votes
0
answers
343
views
Can KL divergence go to 0, but $E[\log(p/q)^2]$ diverge in certain cases?
Let $p(x)$ be a fixed distribution over a discrete space.
Let $A, C > 0$ be constants.
Let $\epsilon > 0$. Can we find an example of a distribution
$q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\...
- 1
0
votes
0
answers
138
views
Why do I not use post hoc tests with a 2 x 2 factorial?
I know this is an obvious answer. I am probably over thinking what I'm doing, but I cannot recall. Does it have to do with not having enough variables to compare the various means?
0
votes
0
answers
362
views
Gradient of the energy functional in $H^{1,2}$-norm
I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
- 2,935
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votes
0
answers
320
views
A result about Fredholm operator
When I read the article "Index Theory" in Handbook of global analysis, I meet a result as below(Corollary 2.13):
If every $F_0\in \mathcal {F}(H_1,H_2)$, there is an open neighborhood $U_0\subseteq \...
- 381
0
votes
3
answers
164
views
Transforming to uniform numbers
Hi
I have a time series of probabilites, vector X
I need to convert the probabilites to uniform numbers.
As I understand it if I put the series into the cdf the output is thus uniform.
The problem ...
- 3
0
votes
0
answers
574
views
What's the expected number of iterations for this process?
Each step of the process consists of choosing a random integer between 1 and the last number chosen this way. On average, how long does it take to obtain "1" as a result of this process for any given ...
- 25
0
votes
1
answer
226
views
Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra
Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of $L^{1}(\...
- 175
0
votes
1
answer
440
views
Variation on Fatou's lemma for Sobolev norms
Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions
$$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$
If I am not ...
- 1
0
votes
1
answer
221
views
Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
- 1
0
votes
1
answer
377
views
Robust entropy-like measure for analyzing uncertainity
I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...
- 5
-1
votes
1
answer
354
views
The grail of functional analysis?
Does $g\in C([0,1],[0,1])=A$ exist such that $\{g^n ,n\in\mathbb N\}$ is dense in $A$ provided with the uniform norm?
with $g^2=g \circ g $
If we can find $g$ then $F$ a closed of $A$, $id \in F$ ...
- 4,074
-1
votes
6
answers
2k
views
Chances to win an election
Let's say that tomorrow national president election is held. A poll asks 1100 persons which of the two candidates, A or B, will he or she will vote. 750 say will vote A, and 250 say will vote B. What ...
- 330
-1
votes
2
answers
409
views
$X$ is Polish and $N$ is countable. Is $N^X$ Polish? [closed]
$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.
$X^N$ is the collection of all mappings from $N$ to $X$. It is ...
- 263
-1
votes
3
answers
3k
views
Monte Carlo method and possible applications to computer poker?
I want to do something about ”games of incomplete information“，like "Computer poker program".I know,Albert university(in canada) have do a lot of things to that field,they write a program called: "...
- 11
-1
votes
3
answers
215
views
Proving the uniform distribution maximizes the expected value of the product of a random draw of $m$ elements from discrete distribution
Say I have a discrete probability distribution $p_i$, so $0 \le p_i \le 1$ and $\sum_i{p_i}=1$. We sample $m > 1$ draws $D$ from this distribution proportional to $p_i$ with replacement, and ...
- 109
-1
votes
2
answers
251
views
$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
- 1
-1
votes
2
answers
217
views
Expected number of balls left out when choosing $n$ times from $n$ balls
I am given $n$ balls. For $n$ times, I pick one of them with uniform probability and put it back after picking it.
Let $U$ be the number of balls I have never picked, so $U\in \{0,\ldots,n-1\}$. We ...
- 48.9k
-1
votes
2
answers
869
views
How can I calculate the expected ranking of a competitor from the probabilities of each competitor reaching first place?
Say I have several competitors contending over some prize. I know the probabilities that any particular one of them will win the prize. It is assumed that the competitors all want to achieve the ...
-1
votes
2
answers
232
views
Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$
Suppose $f \colon [0, 1] \to \mathbb{R}$ is continuously differentiable, and satisfies $f(0) = f(1) = 0$ and $\|f'\|_2 \leq 1$.
I am wondering if it there is a constant $C > 0$ such that for all ...
- 420
-1
votes
1
answer
114
views
Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]
Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
- 217
-1
votes
2
answers
510
views
inequivalent norms [closed]
I am thinking about the following question:
Let $X$ be a Banach space, say separable, e.g., $l_p$ or $c_0$.
When can I say that there exist inequivalent complete norms on $X$?
- 617
-1
votes
1
answer
227
views
Solving the integral identity $ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx. $ [closed]
We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.
Definition: Let $f(x)$ be integrable in $(a,b)$. If ...
- 5,470
-1
votes
2
answers
440
views
$\langle X\rangle_t = t$
Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^d$ and $X_t = |B_t|$. What is the easiest way to see that$$\langle X\rangle_t = t?$$I need this result for a simulation I am running...
- 93
-1
votes
1
answer
149
views
Necessity of Lebegue's convergence criterion? [closed]
the well-known Lebesgue’s dominated convergence theorem states that pointwise convergence of a sequence of functions implies convergence of the sequence of integrals if an integrable function ...
-1
votes
1
answer
139
views
$L^1$ convergence
Setting
For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
- 125
-1
votes
2
answers
407
views
Conditional expectation: commuting integration and supremum
Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
- 109
-1
votes
1
answer
200
views
Is the unordered sum of measurable functions measurable?
Let $E$ be a normed $\mathbb R$-vector space and $I$ be a nonempty set. Remember that $(x_i)_{i\in I}\subseteq E$ is called summable if there is a $x\in E$ such that for all $\varepsilon>0$, there ...
- 167
-1
votes
1
answer
259
views
Which type of convergence for this sequence of random variables? [closed]
Suppose that $X_1,X_2...$ is a sequence of non-negative real random variables. I have that $\mathbb{E}(X_i^2) \to 0$ as $i \to +\infty$, therefore my sequence converges at least in distribution to the ...
- 686
-1
votes
1
answer
108
views
Are local diffeomorphisms Fredholm maps with index zero? [closed]
Does this statement correct? if it does how we can prove it. In Banach spaces
a map is local diffeomorphism if and only if it is a Fredholm map of index zero with no critical points?
- 71
-1
votes
2
answers
641
views
Invariance of spectrum under conjugation
Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
- 677
-1
votes
2
answers
466
views
What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [closed]
How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can satisfy....
- 1,129
-1
votes
1
answer
163
views
Is it true that if a random vector has independent coordinates each bounded by $1$ then $P[ \|X\| \leq \epsilon\sqrt{n}] \leq (C\epsilon)^{n}$?
I'm studying Vershynin's well-written book on "High Dimensional Probability" and the third chapter on concentration of random vectors.
Exercise 3.1.7 from the book is the following.
Let $X =...
- 401
-1
votes
1
answer
223
views
Why do we define independence for zero-probability events? [closed]
I am learning about probability and the definition of pairwise independence is given as $P(AB) = P(A)P(B)$. My textbook motivates this definition as one to capture the intuition where the knowledge of ...
- 9
-1
votes
1
answer
175
views
A strange probability inequality
I need help to understand the following :
For any non-negative random variable $\zeta$: $\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)\leq\mathbb{E}(\zeta)\leq 1+\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)...
- 1
-1
votes
1
answer
88
views
Can we say that: $ \sum_{n\geq 1}{\frac{1}{n}(f_n(\omega)-g_n(\omega))}<\infty\qquad a.e $
Let $(\Omega,\mathcal{A},\mu)$ be a finite mesure space, and $\{f_n\}$ and $\{g_n\}$ two $L^1$-bounded sequences, such that :
$$
\sum_{n\geq 1}{\frac{1}{n}(F_n(f_n)(\omega)-g_n(\omega))}<\infty\...
-1
votes
1
answer
153
views
$\ell^q$ analog of square function
It is a classical result in harmonic analysis that
$$ \|\|P_kf\|_{\ell^2_k}\|_{L^p_x}\approx\|f\|_{L^p} $$
for $p\in(1,\infty)$, where $P_k$ is the Littlewood-Paley decomposition onto frquency $\...
- 5,169
-1
votes
1
answer
332
views
Expectation where linearity does not hold
We have four random variables say W,X,Y,Z where W and X has the same distribution and Y, Z also has the same distribution. Bad news is EX and EY may not exist but E(W+Z) is zero. Could we conclude ...
-1
votes
2
answers
407
views
Almost isometric subspaces of $\ell_p$
1) Given $p\in (1,\infty)$.
2) Let us fix two, non-isometric subspaces $X,Y\subseteq \ell_p$ isomorphic to $\ell_p$.
3) Are there an $\varepsilon\in (0,1)$ and an isomorphism $S\colon X\to Y$ such ...
- 537
-1
votes
1
answer
338
views
about Function of Random variables [closed]
Hello,
I am studying random variables.
Question is this:
if rv X & a function g is known, what is the pdf of random variable Y = g(x)?
in the textbook answer is explained as follows.
P[y ≤ Y ≤...
- 117
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 21
-1
votes
1
answer
168
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349