All Questions
9,497 questions
-1
votes
1
answer
103
views
How to solve MILP problem on several linear subspaces
I have a set of close mixed-integer programming problems. More exactly, all the problems share the same set of (binary and continuous) variables, the same set of linear inequality constraints, and the ...
- 39
-1
votes
1
answer
396
views
Convergence of Radon-Nikodým derivative
Imagine we have a sequence of finite measures $\nu_n << \mu_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have ...
- 183
-1
votes
1
answer
138
views
On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components
Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.
Now, I was wondering ...
- 1,465
-1
votes
1
answer
96
views
On bounding a certain discrepancy between probability distributions on the symmetric group
Disclaimer. This is a follow up to a question I asked and answered on SE https://math.stackexchange.com/q/3579311/168758. The question was about upper-bounds. Here I'm interested in lower bounds, and ...
- 6,853
-1
votes
1
answer
83
views
Convergence in mean and convergence in distribution
Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that
$$
0< ...
- 411
-1
votes
1
answer
370
views
What's the probability of two independent events in time domain?
Suppose there are two independent events A and B. The probability that A or ...
- 29
-1
votes
1
answer
88
views
sparse data fitting problem [closed]
I am a new learner of optimization, and I am confused by the question below, (how to change a 0-norm constrain into binary and linear constrain ?)
Given a sparse data fitting problem:
$ minimize \...
- 1
-1
votes
1
answer
199
views
How can one quantify the convergence of relative frequency to probability?
I have already asked this on stackexchange but did not get any answer.
Say I run a simple Bernoulli trial a number of times and compute the relative frequency for success. Clearly the relative ...
-1
votes
1
answer
159
views
Bounding difference of two mutual information with different distributions on random variables
Let $A$, $B$ and $C$ be three random variables and $p_{A,B,C}=p_Ap_Bp_{C|A,B}$ and $q_{A,B,C}=p_Aq_{B|A}p_{C|A,B}$ be two distributions on them. Then, we can conclude that?
\begin{align*}
\lvert I_p(A,...
- 287
-1
votes
1
answer
519
views
Poisson kernel is the Cauchy distribution, reference?
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
-1
votes
1
answer
364
views
Basketball shots and stopping rule [closed]
Moved over from StackExchange.
You are taken to play a basketball game where you can shoot basketballs at n slots using a machine that is equally likely to shoot the balls into those n slots. You can ...
- 1
-1
votes
1
answer
75
views
Finiteness of "novel variance" from a kernel on a compact space [closed]
Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
- 8,502
-1
votes
1
answer
172
views
Multiplicative version of Novikov inequality for Ito integral
It is clear that Ito isometry
$E(∫^t_0fdW)^2=E(∫^t_0f^2dt)$
can be written in the multiplicative form as
$E(∫^t_0fdW\cdot∫^t_0gdW)=E(∫^t_0f⋅gdt).$
Is it possible to obtain the multiplicative version ...
-1
votes
1
answer
259
views
Absolute continuity of probabilities on Polish spaces and open sets. [closed]
On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
- 41
-1
votes
3
answers
304
views
Distribution under operations
Let $X$, $Y$, $Z$, and $W$ be i.i.d. copies of a standard gaussian variable, that is in distribution $\mathcal{N}\left(0,1\right)$, then what is the distribution of $\left|\frac{XY}{Z}-W\right|$?
...
- 65
-1
votes
0
answers
17
views
Estimate the value of the PDF $P(f)$ at the minimal $f_0$ of the random-variable function $f(\boldsymbol{x})$
Let $f(\boldsymbol{x})=f(x_1,x_2,\cdots,x_N)$ with $N>2$ be a real and continuous function and $f(\boldsymbol{x})\ge f_0$ for any $\boldsymbol{x}\in\mathbb{R}^N$. Now let $x_1,x_2,\cdots,x_N$ be ...
- 375
-1
votes
0
answers
44
views
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\|].
$$
I am wondering if it is possible to ...
- 460
-1
votes
0
answers
41
views
Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
-1
votes
0
answers
35
views
Different definition of Feller semi-group
(This is a crosspost of a question on MathStackExchange which did not receive any answer.)
Let $E$ be a locally compact metric space, let $C_0(E)$ be the set of real-valued continuous functions of $E$ ...
-1
votes
0
answers
27
views
Number variance of random points (and deviations for empirical processes)
Let $X_1, X_2, \dots$ be i.i.d. random variables having uniform distribution on $[0,1]$. Write $I_{t,x}$ for the indicator function of an interval of length $x$ with center $t$. Consider
$$
V(N,x) = \...
- 2,197
-1
votes
1
answer
128
views
(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes
Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$,
\begin{equation}
\int_0^te^{-\lambda ...
- 411
-1
votes
1
answer
76
views
Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution
Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
- 287
-1
votes
1
answer
987
views
Random variable as an integral of an indicator function
This answer says that if $X$ is a random variable and $X_+ = \mathrm{max}(0, X)$, then $X_+ = \int_0^\infty I_{\{X > x\}}\mathrm{d}x$. I'd like to know how to derive this starting with $A \in \...
- 115
-1
votes
1
answer
68
views
Bound for an expectation of random matrix with quantized random variable
Let random matrix $\mathbf{X} \in \mathbb{C^{\mathrm{m} \times \mathrm{n}}}$ and random vector $\mathbf{y} \in \mathbb{C^{\mathrm{m} \times 1}}$ are unknown distributed, but their covariance and ...
- 25
-1
votes
1
answer
108
views
Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law?
In this paper, if we denote the $k$ largest eigenvalues by $\lambda_N,\lambda_{n-1},··· ,\lambda_{N-k+1}, $ then for Gaussian ensembles the joint distribution function of rescaled eigenvalues has the ...
- 288
-1
votes
1
answer
297
views
The distribution of the sum of values from a normal and a truncated normal distribution
Using R to extract truncated normal distribution samples and normal distribution samples separately, when they are combined, the image drawn by the hist function is very similar to a normal ...
-1
votes
1
answer
550
views
Lower bound of an expectation
Suppose a random variable $X$ has unit variance i.e. $\sigma^{2} = 1$. Is there a positive constant $c > 0$ such that
$$\mathbb{E}[\ | X - \mathbb{E}[X] | \ ] \ge c $$
My attempt of a solution is ...
- 643
-1
votes
1
answer
74
views
Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail
Let $X_1,\ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,\ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of ...
- 1
-1
votes
2
answers
420
views
How to define probability over graphs?
How can one formally define a random graph variable?
If G is a random graph variable, then any finite graph is a realization of G. Formally a r.v maps the set of outcomes to a measurable space (may be ...
- 1
-1
votes
1
answer
204
views
How to combine estimator with different variances?
Consider independent random variables $X_1,X_2,\ldots,$ that have the same expectation $\mathbb x=\mathbb E[X_1]=\mathbb E[X_2]=\ldots$
Further, assume that we know that $Var[X_i]=\sigma_i^2$.
In the ...
- 127
-1
votes
1
answer
97
views
Bounding $l^0$ norm of random quantity
There are many techniques in high dimensional probability for bounding quantities of the form
$$ \mathbf{E}( \sup_{s \in S} X_s ) $$
where $\{ X_s \}$ are a family of random variables which are not ...
- 455
-1
votes
1
answer
283
views
Lowerbounding expectation value of binomial tail
I'm trying to find a lower bound for the following expression for $q\ge p$:
$$f(q,p,n) := \sum_{v=0}^n \sum_{k=v}^n \binom{n}{v} \binom{n}{k}q^v(1-q)^{n-v}p^k(1-p)^{n-k}.$$
It can be thought of as the ...
- 702
-1
votes
1
answer
113
views
Approximating expectation of exponential of Wishart matrix
I am trying to obtain an Approximating expectation of exponential of Wishart matrix $X (N,N)$ with $\operatorname{rank} (X)=K$defined as:
\begin{align}
J = E[{e^{{v^H}Xv}}]
\end{align}
where $v$ is $...
- 377
-1
votes
1
answer
312
views
expectation of upper quantile proportion
(edited considerably following comments)
We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$...
- 121
-1
votes
1
answer
114
views
Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values
To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
- 723
-1
votes
1
answer
196
views
Distribution of first time a 1D random walk hits n or -n
Let $(\omega_1, \omega_2, \ldots)$ be iid in $\{-1, 1\}$ and $X_k = \sum_{i=1}^k \omega_i$ be a simple one-dimensional random walk.
Let $\tau_n = \min \{i\in\mathbb{N}: |X_i|=n\}$ be the first time ...
- 516
-1
votes
1
answer
1k
views
Expected value of $W_{t_i} W^2_{t_{i+1}}$
I stuck in determining the expected value of the following product
$E[W_{t_i}W_{t_{i+1}}^2]$ where $W_{t_i}$ and $W_{t_{i+1}}$ are Brownian with normal distribution, i.e. $W_{t_i}\sim N(0,t_i)$. I ...
-1
votes
1
answer
76
views
Transforming random variables for having good property?
For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that
\begin{align}
\Omega&\triangleq \{(x,y): A(x,y)=1\},\\
\Lambda&\triangleq \{x: B(x)=1\}.
\end{...
- 287
-1
votes
1
answer
76
views
transformation of two measures on different space
Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$.
Let $\Sigma_U$ be the smallest $\sigma-$ field ...
- 1
-1
votes
1
answer
428
views
Bins and colored balls
Consider $n$ color balls. We throw them as follows. For a given ball $i$, randomly choose $k$ bins; create $k$ 'copies' of the ball (i.e., of the same color of the ball $i$); throw a 'copy ball' into ...
- 367
-1
votes
2
answers
512
views
Deriving the joint distribution of multivariate normal transformed into Bernoulli
Given a covariance matrix $\sum_{ij}$ and a mean vector $\mu$ I have sampled $N$ multivariate normal vectors $Z = (z_1,...z_n)$ My goal is to create a vector of Bernoulli random variables $Y = (y_1,......
-1
votes
1
answer
459
views
Orthogonal decomposition of conditional expectations
Suppose I have a random variable $x$ and a set of conditional distributions on $x$. Here is an example where the conditionals are nested:
$$q_1 := E(x|y_1), \quad q_2 := E(x|y_1,y_2),\quad q_3 := E(x|...
- 427
-1
votes
1
answer
2k
views
Variance of euclidean norm of Gaussian vectors
Let $X$ be a Gaussian vector in dimension $n$, with $0$ mean and covariance identity. Let $A$ be a square matrix of size $n$, and $Y = A X$. Let $N$ be the square of $Y$ euclidean norm: $N = \sum Y_i^...
- 11
-1
votes
1
answer
1k
views
Rank of covariance matrix whose diagonal elements are same [closed]
Suppose A is a covariance matrix whose diagonal elements are same, i.e. $A_{1,1}=A_{2,2}=\cdots=A_{N,N}$, can we conclude that A is full rank?
Suppose the absolute values of the off-diagonal elements ...
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
- 1,649
-1
votes
1
answer
129
views
How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where Y is Binomial(n,p)
How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where $Y$ is Binomial(n,p)? If it is not exactly computable, then are their ways to approximate this qty?
- 1
-2
votes
3
answers
447
views
Determinant of matrix from set {-1, 1} [closed]
Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$?
I have ...
- 45
-2
votes
1
answer
260
views
On Impossible events
Let's consider a continuous random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$.
Is there a meaningful sense in which one could say that for any $x_0:p(x_0)=...
- 181
-2
votes
1
answer
108
views
Generating a binary probability combination function [closed]
I have been trying to develop a function that can combine two probabilities using the rules:
$f(x,y)\in C^\infty (\mathbb{R}^{2})$
$f(x,y)=f(y,x)$
$f(x,1-x)=\frac{1}{2}$
$f(1-x,1-y)=1-f(x,y)$
$f(0,x)=...
-2
votes
1
answer
282
views
Does convergence in probability implies L^1 convergence in probability density function, for bounded random variables?
Let $X_1,X_2,\cdots$ and $Y$ be random variables on $[0,1]$ with smooth density functions $f_1,f_2\cdots$ and $f$. Suppose $X_n\to Y$ in probability. Can we get some convergence of the density ...
