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.
8,667
questions
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Have some works by Émile Borel ever been translated from French to English or another foreign language?
I plan to submit a couple of questions around Émile Borel's works in probability theory to MO.
In this scope, I'd like to know if the following works have ever been translated from French to English ...
- 1,016
1
vote
1
answer
445
views
Weighted sum of i.i.d. random variables
Suppose you have a positive sequence $X_1,X_2,\dots$ of i.i.d. random variables with the property that
$$
\mathbb{E}[\log(X_1)]<\infty.
$$
Is it true that
$$
\limsup_{n\to\infty} e^{-n}\sum_{k=1}^...
- 491
5
votes
1
answer
388
views
Conditions for existence of dominating $\sigma$-finite measure for all conditional distributions
Suppose $X$ and $Y$ are two real-valued random variables with a specified joint probability distribution $P_{X,Y}.$ I wish to determine if there is a $\sigma$-finite measure $\mu$ on the real line ...
- 1,269
2
votes
1
answer
279
views
Measurability of integrals with respect to different measures
Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
- 395
2
votes
1
answer
240
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Fell topology versus vague topology for representing random sets
I'm trying to better understand the consequences of representing a random set as a
Random element in the space of locally finite closed sets under the Borel sigma algebra generated by the Fell ...
- 321
3
votes
0
answers
775
views
Total variation, Wasserstein, and Prokhorov metrics on countably infinite discrete spaces
Total variation, Wasserstein, and Prokhorov generate the same topology on the space of probability measures on a finite and discrete space.
I'm curious about a countably infinite space. When do ...
- 321
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answers
115
views
Influence of independent variables on boolean functions?
Suppose a simple connected graph $G$ where its vertices are assumed to be independent. An event with uncertainty corresponds to each vertex. My instructor guides me that even though the vertices (...
- 143
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481
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Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question
This is a prequel to my question:
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Clearly my ...
- 1,016
2
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0
answers
79
views
Modify Process to a Semimartingale
The original post is from mathstackexchange
According to some difficulties, i decided to ask here again.
Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a ...
- 247
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votes
4
answers
561
views
How to generalize normal number theorem
The Borel number theorem states that with respect to Lebesgue measure, almost all real numbers are normal numbers. It is sometimes stated in the context of the compact interval $[0,1]$, where one ...
- 121
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229
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Self-adjusting random walk
Let $X_t$ be a random process such that
\begin{eqnarray}
X_1 &=& 0\\
X_t &=& X_{t-1} + \left\{\begin{array}{ll}
A_t, & X_{t-1} \geq 0\\
B_t, & X_{t-1} < 0
\end{array}\...
- 93
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votes
1
answer
361
views
Limit theorem : reproduce a proof with an adaption from discrete to continuous time
Im considering Theorem 5.2.2 in M. Sørensen "Exponential Families of stochastic processes".
The setup is as follows:
We have a Levy-Process $X_t$ fullfilling the CLT
\begin{align}
\sqrt{t}(X_t/t-E(...
- 247
2
votes
0
answers
84
views
Generalizing the law of large numbers to multiple sets of samples
The law of large numbers says that if I sample $n$ points independently from a probability density function $f$, then the number of points lying in a neighborhood of a point $x$ with area $\epsilon$ ...
- 3,929
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votes
2
answers
200
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Expected number of changes in the sign of a rolling sum of independent normal variables
Imagine we define $Y(t+n)=
X(t+1)+.....+X(t+n)$ where $X(i)$ is an independent normal (i.e. everyday we remove the starting observation and we add a new one). We have $n$ consecutive observations of $...
1
vote
1
answer
739
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Restricted Isometry Property for Discrete Fourier Transform Matrix
I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in \mathbb{C}^{d\...
- 1,127
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votes
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views
Literature on transformed Gaussian matrices
I am considering real $n$-by-$m$ matrices of the following type:
$$
M=SM^\prime,\\
M^\prime_{ij}\sim^{iid}N(0,1).
$$
Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
- 121
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votes
1
answer
394
views
An inequality on the simplex involving $x^x$
Is anything known about the behavior of the function $$f(x)=\prod_{i=1}^n x_i^{x_i}$$ on the standard simplex, i.e. the set $\{x\in\mathbb{R}^n:\sum_{i=1}^n x_i=1, x_i\geq0\}$? I ask because I have ...
- 93
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vote
0
answers
83
views
BM hitting times with exponential killing process
Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point $p_s=(x_s,y_s,z_s)$ . BM starts at point $p_0=(x_0,y_0,z_0)$ and when it riches the ...
- 11
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votes
1
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347
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Bounds for the sum of some random variables, in terms of their cdfs
I have three independent non-negative random variables $X_1$, $X_2$, and $X_3$, and I do not have their density functions, but I do have a decent upper bound for their cdfs. In other words, I have ...
- 3,929
6
votes
0
answers
103
views
Random pro-p groups via iterated uniformly random central extensions
Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:
We want to construct an inverse system
$$\cdots \xrightarrow{\alpha_i} G_i \...
- 381
6
votes
1
answer
128
views
Choosing a sample based on where the density function is highest
Is there a name for the following process?
Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. ...
- 3,929
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votes
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85
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A question about probabilistic graphical models
Say one is given a probabilistic graphical model and a cut of the underlying graph. Do we know any statements about when and how can one or many of the marginals (of the sources) or the conditionals (...
- 2,146
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votes
1
answer
135
views
Is there a closed form expression for $E(X e^{-\mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$?
Is there any closed form expression for $E(X e^{- \mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? If not, is there any tight upper bound for this quantity? Any idea how to proceed?
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votes
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answers
155
views
Growth of inner products between two random vectors on the sparse hypercube
We define the $s$-sparse hypercube in $\mathbb{R}^d$ as
\begin{align}
\mathbb{H}_s = \bigl \{ {\bf{v}} \in \{ -1, 0 , 1\}^d \colon \| {\bf{v}} \|_0 = s \bigr\},
\end{align}
where $ \| {\bf v} \|_0 $ ...
- 1,127
3
votes
0
answers
122
views
Batched coupon collector with quota
Assume that you draw coupons uniformly at random from a collection of $n$ coupons and you want to collect $m_i$ coupons of type $i$. This is referred to as the coupon collector with quota (http://www....
- 552
4
votes
2
answers
235
views
The necessary sufficient condition for recurrence of a Markovian random walk
Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk.
I want to figure out the necessary ...
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Famous results about the value of a given limit assuming it exists
Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
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votes
1
answer
93
views
Proving maximal entropy [closed]
It is quite easy to prove that
$$H(S) \leq \log_2(|A|),$$
where $A$ is the number of events, using the Jensen inequality
$$H(S) = E_S[\log_2(\frac{1}{P_S(s)})]\leq \log_2(E_S[(\frac{1}{P_S(s)})]) =...
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votes
4
answers
637
views
Expected value of Bernoulli quadratic forms
Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
- 363
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votes
1
answer
201
views
Sharpened Pinsker inequality for special case
Let $B(p)$ denote the Bernoulli distribution over $\{0,1\}$ and $B(p)^n$ the corresponding product distribution over $\{0,1\}^n$. For $n>1$ and $0<x<1$, define
$$P_n(x):=B(\frac12+\frac x2)^n$...
- 6,061
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votes
1
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541
views
Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]
I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying
$$\int_{\mathbb R}xd\...
- 1,837
2
votes
1
answer
336
views
Integration against Borel measures on compact Hausdorff spaces
I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof.
Suppose that $X$ ...
- 165
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votes
0
answers
257
views
Throwing darts at a barn and putting a bullseye around them in higher dimensions
Let $X \in \mathbb R^d$ be a large domain (a ball of radius $r$ for $r$ large should suffice)
Let $B$ be a ball of radius $1$.
Consider the ratio
$$ \frac{ \left| \left\{ x_1,\dots,x_n \in X \mid ...
- 137k
19
votes
1
answer
436
views
Precise estimate for probability an $n$-point set has diameter smaller than $1$
This question was inspired by an earlier question that I answered but would like a more precise bound for.
Consider random points $x_1, \dots, x_n$ in the unit ball in $\mathbb R^d$, uniformly and ...
- 137k
5
votes
0
answers
116
views
L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension
For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator:
$M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$
(...
- 3,970
2
votes
1
answer
246
views
Question on Wiener processes not hitting 0
Let $W_t$ be a standard Wiener process, and $0\leq a < b$. Let $\hat{W}_t:=W_{a+t}-W_a$. Then $\hat{W}_t$ is also a standard Wiener process. I think that the following should be true:
$$\mathbb P\...
- 121
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votes
1
answer
543
views
Liouville property - a very basic question
Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...
- 4,695
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vote
0
answers
122
views
Alternative to generic chaining bounds for a particular family of stochastic processes
Generic chaining provides a general but rather abstract framework to bound suprema of stochastic processes. In many applications, however, we know more about the expression of the stochastic process. ...
- 215
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votes
0
answers
107
views
Probabilistic distribution of sandpile model type
Let $G=(V,E)$ be a connected graph. Assume that $m\leqslant |V|$ hedgehogs sit in the vertices of $G$. If there are $r\geqslant 2$ hedgehogs in the same vertex $v\in V$, one of them goes to a randomly ...
- 103k
-1
votes
1
answer
184
views
Equal probability of having even/odd number of ones in many Bernoulli trials with different probabilities? [closed]
This problem has probably been solved somewhere but I could not find it. We have $n$ Bernoulli random trials $X_i$ with different occurrence probabilities, $\mathrm{Pr}[X_i=1]=p_i>p_{\min}>0$ ...
- 297
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vote
0
answers
87
views
How to estimate the size of balanced biclique in random bipartite graph?
We have a random bipartite graph $G=(V,U,E)$ and $|V|=|U|=n$, in which any vertex pair $<v,u>$ ($v\in V$,$u\in U$) exists an edge with probability $p$. A balanced bipartite complete graph is a ...
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votes
2
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views
Shortest path through $n^{1/3}$ points out of $n$
Say I sample $n$ points uniformly at random in the unit cube in $\mathbb{R}^3$, and then I look for the shortest path through $n^{1/3}$ of those points (rounding up, say). What happens to the length ...
- 141
1
vote
1
answer
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views
How to calculate the expected value of complex-valued random variable? [closed]
Suppose $\theta_1,\theta_2,\cdots, \theta_n$ are independent and identically distributed (i.i.d.) real-valued random variables and here we specifically consider the uniform distribution in the ...
- 19
2
votes
1
answer
294
views
About Renyi entropy
If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = \frac{...
- 2,146
2
votes
1
answer
637
views
Averages of vector inner products over the Haar measure
Consider arbitrary unit vectors $w,x,y,z \in \mathbb{C}^d$. Is there an explicit formula for what this average is?
$$
\int \mathrm{Tr}( \psi \psi^* \, \, w x^* \,\, \psi \psi^* \,\, y z^*) d\psi
$$
...
- 1,909
1
vote
1
answer
1k
views
Limit (convergence) of stopping times
Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
- 1,837
7
votes
1
answer
668
views
Tightness and Functional Analysis
Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an ...
- 2,283
1
vote
0
answers
102
views
What is meant by local time of BM on the boundary $\partial D$?
I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
- 270
1
vote
1
answer
314
views
Averaged geometric series with floor function
Given a value $p\in[0,1]$ (a probability of occurrence), I would like to bound the following expression:
$$ s\frac{1-(1-p)^{k+1}}{p(k+1)} + (1-s)\frac{1-(1-p)^{k}}{pk},\ \ \ \text{where $k=\lfloor 1/...
3
votes
0
answers
155
views
Question about martin boundaries of random walks induced on transient subgroups
Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and
consider a random walk given by a measure $\mu$.
Assume the measure is symmetric, finitely generated, and the support of
$\...
- 2,490