Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,024 questions
0
votes
1
answer
284
views
Is there a "smooth Kantorovich-Rubinstein duality" for Wasserstein distances on smooth/Euclidean space?
Let $X$ be a compact metric space, and fix an arbitrary point $x_\ast \in X$. By the Kantorovich-Rubinstein duality theorem, the $1$-Wasserstein metric $W_1$ on the set of Borel probability measures ...
- 708
0
votes
1
answer
128
views
Non-linear diffusion on networks
The diffusion equation with constant diffusion $D$ can be represented as:
\begin{equation}
\frac{\partial \phi(r, t)}{\partial t}=D \Delta \phi(r, t)
\end{equation}
where
$\Delta$ is the Laplace ...
- 117
0
votes
1
answer
643
views
A good approximation for collision probability between (two) sets of random variables
We face many places to find the collision probability of two sets (or more) in my case the cryptographic hash functions. We can formalize as;
Given two sets of random variables $\mathbf{A}$ and $\...
- 115
0
votes
1
answer
204
views
Bernoulli random matrix and rotationally-invariant ensemble [closed]
I am recently reading about random matrix theory in engineering applications. I come up with the following question. I have been trying to find any references but it doesn't help. Hopefully, anyone ...
- 87
0
votes
2
answers
201
views
Couplings as generalized functions
I've been casually reading about optimal transport, and I was intrigued by the Wasserstein metric, in which we define the distance between two measures $\mu$ and $\nu$ on a metric space $X$ by
$$
W_p(\...
- 123
0
votes
1
answer
147
views
Trying to prove an inequality (looks similar to entropy)
I'm trying to prove the following inequality (or something similar, up to a constant factor in either side of the inequality):
$$k\cdot\sum_{i=1}^{k}x_{i}\cdot\ln\left(x_{i}\right)\geq\sum_{i=1}^{k}x_{...
0
votes
1
answer
66
views
Bounding parameter satisfying a collection of inequalities
I have a set of equations with some inequality constraints that I expect generally does not have a unique solution.
The equations take the form below:
$$\alpha/N+(1-\alpha)x_1=a_1$$
$$\alpha/N+(1-\...
- 135
0
votes
2
answers
341
views
Conditions for existence of a distribution with full support
Consider a $6\times 1$ continuous random vector
$$
\eta\equiv (\eta_1,\eta_2,..., \eta_6)
$$
satisfying the following property:
$$
\underbrace{\begin{pmatrix}
\eta_1\\
\eta_2\\
\eta_3
\end{pmatrix}}_{\...
- 108
0
votes
1
answer
154
views
Q-Gaussian processes and probability space
It is well known that for isonormal Gaussian processes, one can decompose the space $L^2(\Omega)$ into Wiener chaos so that the space $L^2(\Omega)$ is isomorphic to the direct sum of the Wiener chaos, ...
- 321
0
votes
1
answer
138
views
Do measure-valued dynamical systems correspond to marginals of Markov processes?
Let $(\mu_n)_{n=1}^{\infty}$ be a sequence in $\mathcal{P}_1(X)$ for some compact metric space $(X,d)$. Suppose that there is a weakly-continuous function $F:\mathcal{P}_1(X)\rightarrow \mathcal{P}_1(...
- 5,405
0
votes
1
answer
217
views
On independence of multiples of $\mathbb Z_p$
This is a rewording in combinatorial language of a question posed on another forum. The original was posed as a probabilistic problem.
Problem set up:
Consider for a fixed prime $p$, the ...
- 6,223
0
votes
1
answer
134
views
Integral bound for square of log derivative
I am currently facing the following problem:
Given a polynomial $f(x) = \sum_{s \in S_f} u_s x^s$, $f(0)\neq 0$, $\lvert S_f \rvert \leq t$ (i.e. $f$ is $t$-sparse) with $u_s$ coming as samples from i....
- 13
0
votes
1
answer
288
views
Random sampling from modified Erlang distribution [closed]
I am tasked with randomly sampling from the following probability density function, which is a modified Erlang Function:
$$f(k,q,\nu)=\frac{(k q)^{k-1}}{[(k-1) !]^{v}} \quad \text { with } \quad q \...
- 111
0
votes
2
answers
452
views
Expected value of Taylor series with central moments of binomial variate
I want to understand this entry, but do not understand how the $\mathcal{O}\left(\frac{1}{n^2}\right)$ in the accepted answer comes into play.
I reproduce the question here: We have $x \sim \mathrm{...
- 89
0
votes
1
answer
271
views
Associativity rule for integration against fractional Brownian motion
In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...
- 103
0
votes
1
answer
77
views
A question concerning terminal time
A terminal time $\tau$ is a stopping time satisfying $$\omega \in \{\tau(\omega) > t\} \text{ implies that } \tau(\omega) = t + \tau(\theta_t\omega), $$ for all $t\ge 0$. Here $\theta_t$ is the ...
- 622
0
votes
2
answers
80
views
Name of distribution of the parameter of a Poissonian
Consider a Poisson process $\hat{n}$ with with parameter $t$ and distribution
$$f_t(n) = e^{-t} \frac{t^n}{n!}$$
Now instead suppose to have a random variable $\hat{t} \in \mathbb{R}^+$ whose ...
- 503
0
votes
2
answers
266
views
Last crossing of a line by a random walk
Let $X_1, X_2, ...$ be i.i.d. random variables, $\mathbb{E} X_1 > 0$, and let $S_n = \sum\limits _{i = 1} ^n X_i$. Define $\tau = \max \{n \in \mathbb{N}: S_n \leq 0 \}$ with the convention $\tau =...
- 724
0
votes
1
answer
294
views
Joint distribution of random Fourier coefficients
Consider choosing a Boolean function $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ uniformly at random from the set of all Boolean functions and consider the random variable $\left(\hat f(z_{1}), \hat f(z_{...
0
votes
1
answer
582
views
Integrability of $\int \log(f(x)) f(x) dx$ for a probability density function $f$
I am looking for weak conditions when a probability density function $f$ on $\mathbb{R}^d$ has a finite integral
$$
\int_{\mathbb{R}^d} \log(f(x)) f(x) dx.
$$
Any references would be appreciated.
- 3
0
votes
1
answer
124
views
Question on limit in probability of the ratio of max to min of 2 sequences of non-ive, continuous iid random variables with support $[0, \infty).$
For each $ m \ge 1$, let $X_m$ and $Y_m$ be two non-negative iid random variables with the same distribution. (The distributions of $X_m$ may change with different $m$.)
**Assume that their support of ...
- 1,467
0
votes
1
answer
260
views
Entropy of a refinement of a partition
We consider a probability space $(X, B, \mu)$. Let $\alpha$ and $\beta$ be countable partitions of X. We suppose $\beta$ is a refinement of $\alpha$, ie that every set in $\alpha$ is a union of sets ...
- 41
0
votes
1
answer
115
views
Average over spheres finite
Let $X_1,...,X_N$ be random variables that are iid with the uniform distribution over $\mathbb S^n.$
I am curious how to see that $f(X_1,..,X_N):=\left \lvert \sum_{i=1}^N X_i \right\rvert^{-1}$ has ...
- 124
0
votes
1
answer
809
views
Concentration of $\ell_2$ norm of a vector sampled from a distribution
Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.
I'm ...
- 61
0
votes
1
answer
267
views
Large deviation for Brownian occupation time
I am asking for reference about the large deviation principle (LDP) for the occupation time of a Brownian motion/bridge. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth and compactly supported. My ...
- 103
0
votes
1
answer
195
views
Gaussian integral $\int_X \|x\|_X^2 \mu(dx)$ in Banach space
For a centered Gaussian measure $\mu$ on a Hilbert space $X$, it is known that
$$\int_X \|x\|^2 \mu(dx) = tr(Q)$$ where $Q$ is the covariance operator. Is there a similar version for Gaussian measures ...
- 375
0
votes
1
answer
179
views
How to show $\max_{1\leq i\leq n}(X_i+Y_1)\preceq \max_{1\leq i\leq n}(X_i+Y_i)$?
Let two collections of random variables $\{X_i\}$ and $\{Y_i\}$ be independent and let $\{Y_i\}$ be i.i.d. Then
$$\max_{1\leq i\leq n}(X_i+Y_1)\preceq \max_{1\leq i\leq n}(X_i+Y_i).$$
where $\...
- 288
0
votes
1
answer
210
views
Distribution of the direction of Gaussian random variable
Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...
- 1,069
0
votes
1
answer
199
views
Is this probability inequality true?
This question may be simple, though I'm not managing to find an answer. Let $X$ and $Y$ be two dependent random vectors in in $\mathbb{R}^d$, with joint probability density $\mu(x,y)$ (with respect to ...
- 195
0
votes
1
answer
121
views
$\sum_{n=1}^{\infty}{\frac{1}{n^{1+\epsilon}}\mathbb{E}\big((|X_n|\mathbb{1}_{|X_n|\leq n})^{1+\epsilon}\big)}<\infty,~~\forall\epsilon>0 $
Let $(E,\mathcal{A},\mathbb{P})$ be a probability space $\{X_n\}$ be a sequence of random variable, such that:
$$
(1)~.~~~\sup_n\mathbb E (|X_n|)<\infty\Rightarrow
$$
$$
(2)~.~~~\dfrac{M_j}{2}<...
- 115
0
votes
1
answer
1k
views
Expectation of inverse of random matrices
Assume that $\mathbf{X}$ is a random positive-definite matrix. Then, is there any upper or lower bound on the expectation of the following expression
$$\mathbb{E}[\mathbf{X}^{-1}]-\alpha\mathbb{E}[\...
- 287
0
votes
1
answer
153
views
Probability distribution of random products of elements of a generating set of a finite non-abelian group
Let $G$ be a finite non-abelian group, and consider a choice of $N$ distinct elements $g_{0},g_{1},\ldots,g_{N-1}\in G$ that generate $G$. Now, let $t$ be an arbitrary positive integer, and let $d_{1},...
- 1,284
0
votes
1
answer
181
views
The mean of a running maximum
Suppose $𝑊$ is a one-dimensional standard Brownian motion defined on some probability space $(\Omega, \mathcal F, P)$ and let $𝑋(𝑡):=\exp\{𝑊(𝑡)−\frac{1}{2}𝑡−\frac{1}{𝑡+1}\}$ for $𝑡\ge 0$. Note ...
- 622
0
votes
1
answer
101
views
Question of expected number of consecutive coin flip with increasing bias [closed]
This is a question I found on the book and I don't know how to tackle it. Thanks to any help or hint in advance.
I have a coin that, I could get the head 100% at the first flip, $\frac{1}{3}$ at the ...
- 11
0
votes
1
answer
163
views
Sphere inversion in Riesz potential
I am reading the paper: ``ON THE DISTRIBUTION OF FIRST HITS FOR THE SYMMETRIC STABLE PROCESSES" by Blumenthal, Getoor and Ray, (Trans. Amer. Math. Soc. 99 (1961), 540-554).
On page 546, the authors ...
- 3
0
votes
2
answers
244
views
Spectrum of a Markov kernel acting on $L^2$
Let $P$ be a Markov kernel on a measurable space $(E,\mathcal E)$ admitting an invariant probability measure $\pi$. $P$ acts on $L^2(\pi)$ via $$Pf:=\int\kappa(\;\cdot\;{\rm d}y)f(y).$$ The invariance ...
- 167
0
votes
1
answer
112
views
PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$ [closed]
How can I find the PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$, $i.e.$, a uniformly distributed r.v.?
My difficulty here is that it involves complex numbers and I don't know ...
0
votes
1
answer
196
views
Coupling between two distributions
Consider $s = \Theta(n^{\delta})$ for a $\delta\in (0,1)$ and let $p\in (0,1)$ with $m = \lfloor pn\rfloor$. Consider the random variable $Y$ which chooses $m$ elements from $\{1,\ldots,n\}$ such that ...
- 49
0
votes
2
answers
306
views
Lower bounds on discrete time finite Markov chains hitting probabilities
I am interested in some general theorems related to lower bounds on discrete time finite Markov chains hitting probabilities (preferably ergodic chains , but not necessarily ), with references . ...
0
votes
1
answer
101
views
If a strong Markov process reaches a Borel set a.s., can it be restarted from that set?
Let $X$ be a strong Markov process on $E$, and $B\in \mathcal B(E)$. Suppose that, for some $x\in E$,
$$
P_x(\exists t\ge0 \text{ such that } X_t\in B)=1.
$$
My question: Does there exist a stopping ...
- 39
0
votes
1
answer
75
views
Expected sum of chosen coordinates in a random subset of a Hamming hypercube
Let $S$ = $\{v_1, v_2, ..., v_n\}$ denote a random subset of a Hamming hypercube of dimension $d$, where $n = |S|$ and $n \leq 2^d$. If $v_i$ = $\langle x^i_1x^i_2... x^i_d\rangle$ for all $i \in [1,n]...
- 11
0
votes
1
answer
143
views
Right tail decay of F distribution [closed]
Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$?
$$\mathbb{P}(X\geq x)$$
what is the order of the above probability as $x\to+\infty$?
- 1,495
0
votes
1
answer
254
views
Does log-concave approximable distribution satisfy transportation-cost inequality?
Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition
$$
\...
- 6,853
0
votes
1
answer
308
views
Berry-Esseen type theorem for Monotonic independence
The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases ...
- 152
0
votes
2
answers
141
views
A simple equation with a normal distribution [closed]
Let $f$ be the distribution of a normal variable $\mathcal{N}(0,1)$, ie
$$ f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$
I have to solve the equation:
$$x+y = f(x) - f(y)$$
I was working with mathematica ...
- 131
0
votes
1
answer
221
views
Non-erasure probability in a loop-erased random walk in three dimensions
Perform a simple random walk $S(0),S(1),S(2)...$ on $\mathbb{Z}^3$, that is $S:\mathbb{N}\to\mathbb{Z}^3$ with $||S(i)-S(i+1)||_1 = 1$ for all $i$. Now let $\Gamma_n$ be the loop-erasure of the first $...
- 191
0
votes
1
answer
369
views
How to derive this change of measure identity in multi-armed bandits?
We have two Bernoulli distributions with success probabilities $\mu_1$ and $\mu_2$. We sample $n$ times from distribution 1 and the sequence we get is $X_1, \ldots, X_n$. Let
$
\hat{kl}_s = \sum_{t=...
- 145
0
votes
1
answer
150
views
Proving an inequation based on binomial distributions
Problem statement
Let $c \in \mathbb{N}$, $n_1 \in \mathbb{N}_0$, and $n_2 \in \mathbb{N}_0$ be integers and $p$ a probability. Furthermore, let $b(m,j,p) = \binom{m}{j}p^j(1-p)^{m-j}$ denote the ...
- 11
0
votes
1
answer
191
views
Asymptotically full stationary process
Let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is asymptotically full if for every increasing sequence of subsets $B_n \subset A^n$ such that $\dfrac{\#B_n}{\#...
- 2,560
0
votes
1
answer
70
views
How can two random variables are continuous infers that their jointly random variable is continuous [closed]
We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable.
But we don't assume that $X$ and $Y$ are independent.
My question is the following:
Is it true that the ...
- 9