All Questions
Tagged with reference-request pr.probability
96 questions
8
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1
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Conditional law as a random measure and convergence of random measures
I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider ...
- 2,560
8
votes
2
answers
562
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When do iterated conditional expectations converge?
Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X$ satisfying $\mathbf{E}[|X|]<\infty$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...
- 1,068
7
votes
0
answers
222
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Projected polar chessboard measure convergence in total variation?
$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set
$$F_n:=\...
- 128k
7
votes
0
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452
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Does the law of a Feller Process on a non-locally-compact Polish space depend continuously on the initial condition (in Skorohod path-space)?
I am sure this is written down somewhere but cannot find it.
Consider a Polish space $E$ and a strong Markov process $(X_t)_{t\ge 0}$ with values in $E$ and cadlag paths. More precisely, we have a ...
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7
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1
answer
391
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Idempotent splitting for Markov kernels
Let $X$ be a standard Borel space and $e : X \to X$ a Markov kernel. Suppose that $e$ is idempotent, that is $e \circ e = e$, or written out using the Chapman-Kolmogorov equation,
$$e(A|x) = \int_X e(...
- 6,406
7
votes
2
answers
3k
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The largest Wasserstein distance to uniform distribution among all probability distributions with uniform marginals
I am looking for the largest Wasserstein distance to the uniform distribution among all probability distributions with uniform marginals.
More specifically, let $\Xi=\{1,2,\ldots,N\}^2$, and let $\nu$...
- 422
7
votes
2
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984
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Brownian motion in $n$ dimensions
Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in \...
- 73
6
votes
1
answer
1k
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About the generating structure of Borel field
This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer.
On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
- 8,071
6
votes
1
answer
196
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Simultaneous simulation of all probability measures on a compact metric space
A well known fact in probability is that a uniform random variable on $[0,1]$ can be used to simulate any other probability distribution on $\mathbb{R}$.
A standard way of doing this is to define, ...
- 4,304
6
votes
3
answers
938
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Uniformly distributed sequence in $\mathbb{R}$
We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and
$$\lim_{N \to \infty} \...
- 61
5
votes
1
answer
263
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Reference request: Urbanik's work on random integrals and Orlicz spaces
Several important papers on Lévy processes are referring to the following paper:
K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces,
Bulletin de l'Académie Polonaise des Sciences, ...
- 2,306
5
votes
1
answer
512
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Concentration inequality for Hilbert space valued random variables
I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
- 115
5
votes
0
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665
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Concentration of sums of random vectors under moment conditions on the marginals
Let $X_1,\dots,X_n\in {\bf R}^d$ be $d$-dimensional iid zero mean random vectors with covariance matrix $\Sigma$. I am interested in tail bounds for the Euclidean norm
$$N_n\equiv \frac{1}{\sqrt{n}}\|...
5
votes
1
answer
2k
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Mathematics research relating to machine learning
What branch/branches of math are most relevant in enhancing machine learning (mostly in terms of practical use as opposed to theoretical/possible use)? Specifically, I want to know about math research ...
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4
votes
1
answer
474
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Concentration inequalities in $\ell_{\infty}$ for sums of iid random ("nice") functions?
I'm looking for "tail-bound-like" inequalities that look like this (I state a specific setting but more general settings are interesting):
Let $D$ be a distribution on a set of "nice" functions $g$:...
- 4,529
4
votes
1
answer
694
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Concurrency related problems in $n$ independent, parallel $M/M/1$ queues
Queueing Model:
Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rate $\lambda$ and service rate $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) ...
- 139
4
votes
1
answer
364
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Reference for multivariate generalised CLT
I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 380
4
votes
2
answers
667
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Convergence (topology) for $\sigma$-finite measures
I'm having much trouble finding literature that addresses the questions which I write below. I was wondering if someone could help me out to understand better, either by providing references or by ...
- 361
4
votes
1
answer
197
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On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
- 128k
4
votes
2
answers
374
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Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
- 657
4
votes
0
answers
188
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Distributions over permutation groups $\mathcal{S}_n$
Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
- 8,071
4
votes
1
answer
311
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Examples of Borel probability measures on the Schwartz function space?
Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions.
Minlos Theorem as ...
- 3,477
4
votes
2
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255
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Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 835
4
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1
answer
863
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Hoeffding's inequality for Hilbert space valued random elements
Suppose that $\mathbb H$ is a separable Hilbert space and $X_1,\ldots,X_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|...
- 423
3
votes
1
answer
476
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distribution discretization
Let $\mu$ be a distribution on $\mathbb{R}^n$. We partition $\mathbb{R}^n$ into small cubes congruent with $[0,\delta)^n$, parallel to the axes. In each cube, pick a point $x$ (for instance, the ...
- 640
3
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2
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271
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For a SDE with smooth transition densities, if every point is "path-accessible", is every positive-measure set probabilistically accessible?
Suppose we have a $C^\infty$ manifold $M$ and $C^\infty$ vector fields $b,\sigma_1,\ldots,\sigma_k$ on $M$, and for convenience define the set of vector fields
$$ \mathcal{S} = \{b,\sigma_1,-\sigma_1,\...
- 708
3
votes
1
answer
131
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Reference request: “A random integral and Orlicz spaces” [duplicate]
I need to find the following paper:
“K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques ...
- 77
3
votes
0
answers
237
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Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k
3
votes
1
answer
253
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Bounds for duplicate finding with limited independence
(This is a follow up to this previous question on math.stackexchange.com.)
Assume a process that samples uniformly at random from the range $[1,\ldots,n]$. I am interested in the time to find a ...
- 33
3
votes
1
answer
181
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How tight is the bound $P(\|X\|^2 \ge t |\langle a,X\rangle|) \ge 1 - t\sqrt{\frac{2}{m-1}}$, where $X \sim N(0, I_m)$ and $\|a\| = 1$?
Let $X$ be a random vector in $\mathbb R^m$ with iid $N(0,1)$ coordinates and let $a$ be a fixed unit vector in $\mathbb R^m$. In another post (SE link here https://math.stackexchange.com/a/3792730/...
- 6,853
3
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2
answers
4k
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Asymptotics of the maximum of binomial random variables
Let $B_i(n,1/2)$ be independent identically distributed binomial random variables. I am interested in the asymptotic growth of the maximum of $n$ such random variables. In https://math.stackexchange....
user32786
3
votes
1
answer
229
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Inequality for difference of consecutive atom probabilities for binomial distribution
Edit: This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its ...
- 2,720
2
votes
1
answer
309
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Upper bound Wasserstein distance by $\chi^2$ distance
Given two random variables $X,Y$ which are both $\mathbb{N}$-valued and have the same expected value (which is some fixed positive constant), and denote their probability mass functions by ${\bf p} = (...
- 730
2
votes
1
answer
154
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Reference Request for Couplings with Conditions
I have two discrete (integer-valued) random variables $A,B$, with $1\le A\le n$ and $1\le B$. A coupling is a joint distribution of $A,B$ with marginal distributions $A,B$. I know there are several ...
- 329
2
votes
1
answer
170
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Law of large numbers for a continuum of Bernoullis
Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
2
votes
1
answer
210
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Prove / disprove: If $1 \le n < N$ and $A$ is an $N \times n$ matrix with iid from $\mathcal N(0,1)$, then $s_\min(A) \ge c\sqrt{N}$ w.p $1-2e^{-N}$
Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) ...
- 6,853
2
votes
2
answers
248
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On an angle distribution of a random linear subspace of a given dimension
$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
- 128k
1
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1
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344
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Is the Borel-Cantelli Lemma applicable here? [duplicate]
Consider $(X_{n})_{n\in\mathbb{N}}$ a sequence of random variables taking values in the set $\mathbb{Z}_{\geq 0}$ where $\mathbb{P}(X_{n} = i) > 0 $ for every $i\in\mathbb{Z}_{\geq0}$ which are ...
- 161
1
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0
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83
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Tracy Widom type results for asymptotic distribution of the $k$-th largest eigenvalue of the sample covariance when $n, p \to \infty$?
Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...
- 1,467
1
vote
1
answer
412
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Exit time estimate for a simple continuous-time random walk
Let $S = (S_t, t \geq 0)$ be a simple one-dimensional continuous-time random walk with total jump rate one, $S_0 = 0$. Denote by $T_k$ the time when $S$ exits the interval $I_k = [-k,k] \cap \...
- 724
1
vote
1
answer
124
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References: error and stability estimates for information projection
$\newcommand\SS{P}\newcommand\TT{Q}$I will call a Gaussian probability measure $\SS$ on $\mathbb{R}^d$ isotropic if its covariance matrix is diagonal with non-vanishing determinant; i.e. $\Sigma_{i,i}&...
- 364
1
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1
answer
97
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A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$?
To get the non-strict version of ...
- 128k
0
votes
1
answer
340
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Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
0
votes
0
answers
115
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Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?
Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
- 1,467
0
votes
0
answers
113
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How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
- 1,677
-6
votes
2
answers
2k
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Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a ...
- 43.2k