All Questions
112 questions
1
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0
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53
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The limit ratio of two Markov Chain Probability
Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
- 303
2
votes
0
answers
84
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Concentration result for self-normalized empirical process
In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....
- 206
0
votes
0
answers
73
views
Asymptotic stochastic ordering for weighted sum of i.i.d. random variables
Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$,
\begin{equation}
a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
- 19
2
votes
1
answer
187
views
Law of iterated logarithm for quadratic variation of Brownian motion
Let $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space supporting a standard Brownian motion $B$. Let $\Pi=\{\pi_n : n \ge 0\}$ denote the sequence of dyadic uniform partitions of the ...
- 41
1
vote
0
answers
112
views
A high probability bound for a Rademacher process
Let $\{x_i(t)\}_{i=1}^n$ be i.i.d. Gaussian processes for $t \in [0, T]$ with
\begin{align*}
\mathbb{E}[x_i(t)] & = 0, \quad \forall i \in [1 : n], \ t \in [0, T] \\
\mathbb{E}[x_i(s) x_i(t)] &...
- 81
1
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0
answers
39
views
Characteristic function of a Dirichlet Process
Suppose $P \sim \text{DP}(\alpha,G) $ where $G \sim N(0,1)$ is the base measure and $\alpha > 0$ is the concentration parameter. The stick breaking representation says that $P$ can be expressed as \...
- 11
0
votes
2
answers
261
views
What mathematical formalism might be used to disprove natural selection, on the basis that there are too many independent genetic parameters? [closed]
I have nagging doubts that the random genetic mutation process of natural selection is sufficient to explain evolution, even when coupled with sexual selection (Darwin proposed that evolution is ...
- 41
4
votes
1
answer
258
views
When is $\prod_{i=0}^\infty (I-x_i x_i^T)=0$ for isotropic Gaussian $x_i$?
Suppose $x_i$ is sampled IID from isotropic zero-centered Gaussian random variable in $d$ dimensions with covariance $\Sigma=c*I$. When is the following true with probability 1?
$$\prod_{i=0}^\infty (...
- 2,252
9
votes
1
answer
723
views
Popular mistakes in probability
$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
0
votes
1
answer
177
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Under which conditions Mean Square Continuity implies Sample Continuity for Gaussian Processes?
First, let us give the setting.
Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process.
By Mean ...
- 141
1
vote
1
answer
93
views
An inequality relating $\ell_1$ distance of input and output of a Markov krnel
Let $K$ be a Markov kernel from $\mathcal{X}$ to $\mathcal{Y}$, i.e., $K(\cdot|x)$ is a probability measure on $\mathcal{Y}$ for all $x\in \mathcal{X}$.
Let $\mu$ and $\nu$ be two probability measures ...
- 1,109
2
votes
1
answer
150
views
Normalized concentration inequality for empirical CDF (iid sum)
Consider the empirical and population CDF,
$$
F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad
F(t) = \mathbb{E} [F_n(t)],
$$
where above $X_1, \dots, X_n$ are iid, real-...
- 420
1
vote
1
answer
199
views
Rademacher complexity for a family of bounded, nondecreasing functions?
Let $\{\phi_k\}_{k=1}^K$ be a family of functions mapping from an interval $[a, b]$ to $[-1, 1]$.
That is, $\phi_k \colon[ a,b] \to [-1, 1]$ are nondecreasing maps on some finite interval $[a, b] \...
- 420
1
vote
1
answer
229
views
Gaussian width of intersection of cube and ball in high-dimensional euclidean space
Let $d$ be a large positive integer and fix $r \ge 0$. Set $S := B_2^n \cap [-r,r]^d$, where $B_2^d$ is the euclidean unit-ball in $\mathbb R^d$. Finally, let $\omega(S)$ be the Gaussian width of $S$, ...
- 6,853
0
votes
1
answer
82
views
WLLN for bootstrap means of stationary ergodic processes?
Setup:$\quad$
Suppose that $(X_n)$ is a stationary ergodic process with $E|X_1|<\infty$.
Given $X^{(n)}=(X_1, \dots, X_n)$, select a standard Efron bootstrap subsample $(X_{n,1}^*, \dots, X_{n,m(n)}...
- 127
1
vote
1
answer
385
views
How fast does this Gaussian random walk move away from the origin?
Suppose $z_i$ are IID zero-centered $d$-dimensional Gaussian random variables with unit-trace covariance $\Sigma$ and $g(z_i)$ is the sum of its components.
Consider the following random walk:
$$x_s=\...
- 2,252
2
votes
0
answers
124
views
Rough path expected signature vs cumulant-generating function / characteristic function
What is the point of using rough path expected signature to characterize the law of а stochastic process when the cumulant generating function is known ($\log\mathbb{E}[e^{i\theta X(t)}]$)?
Since an ...
- 53
3
votes
1
answer
532
views
What is a tensor product of random variables?
I am trying to understand the the following paper https://arxiv.org/pdf/1810.10971.pdf, in particular Example 2:
If $ Y \sim N(0,1)$, the standard normal on $\mathbb{R}$, then
$ \begin{align*} \Big( \...
- 53
1
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0
answers
83
views
Properties of max of many linear combinations of a multivariate normal vector and/or sum of top $k$ elements of a multivariate normal vector
Thank you in advance for your help!
I am interested in studying the following probability:
$$P\big[\max_{H \subset X,|H|=k} \sum_{i \in H} \mathbf{a}_i^T \mathbf{w} \ge 0 \big],$$
where $\mathbf{a}_i$ ...
- 11
1
vote
1
answer
216
views
Rademacher complexity of function class $(x,y) \mapsto 1[|yf(x)-\alpha| \ge \beta]$ in terms of $\alpha$, $\beta$, and Rademacher complexity of $F$
Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$...
- 6,853
1
vote
1
answer
221
views
Large deviation for empirical median
I found this exercise while reading some notes on Large Deviation Principle. This exercise is at the end of the very first chapter, including Cramer's Theorem and essentially nothing more (no Sanov ...
- 445
1
vote
2
answers
277
views
Distribution of interarrival times for a special class of stochastic point processes
I am interested in Poisson-binomial stationary point processes (here on the real line) defined as follows. Let
$t_k=k/\lambda$, with $k\in\mathbb{Z}$ and $\lambda>0$,
$F_s(x)$ be a symmetric, ...
- 3,098
1
vote
1
answer
410
views
Occupation times for two-state Markov processes
Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ ...
2
votes
1
answer
110
views
Lower bound on likelihood of binary outcomes
I am wondering about the following: does there exist a stochastic process $(X_n)_{n \ge 1}$ with values in $\{0,1\}$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ such that for all $n \ge 1$...
- 341
0
votes
1
answer
142
views
Covering number of the conditional distribution function
Suppose $Y$ is a random variable in $\mathbb{R}^d$, and we want to find the covering number
\begin{equation*}
\mathcal{F} = \big\{ F_{Y|W} (y | W) : y \in \mathbb{R}^d \big\}
\end{equation*}
where ...
- 331
0
votes
1
answer
160
views
Probability to cross an envelopp for 1D random walk?
Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence.
I can make an analogy with random walk: let ...
- 5
3
votes
0
answers
98
views
Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix
In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
- 279
1
vote
0
answers
68
views
(Anti-)concentration of gap between largest and second largest component of multivariate random gaussian vector
Let $n$ be a large positive integer and let $Y=(Y_1,\ldots,Y_n)$ be a zero-centered random $n$-dmensional real vector with covariance matrix $\Sigma$, an $n$-by-$n$ positive definite matrix with ...
- 6,853
1
vote
0
answers
49
views
semi-parametric regression
Suppose the observation $(X_1, Y_1), \ldots, (X_n, Y_n)$ satisfies the following semi-parametric model
$$Y_t = m(X_t, \alpha) + \sigma(X_t, \beta) U_t,$$
where $U_t$ is independent with $X_t$ with ...
- 331
2
votes
2
answers
206
views
non-homogeneous counting process
Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$....
- 31
1
vote
0
answers
99
views
Covering number after projection
In these lecture notes on Statistical Learning Theory we find the following definitions for covering numbers:
Definition. Let $(\mathcal{W}, d)$ be a metric space and $\mathcal{F} \subset \mathcal{W}$...
1
vote
1
answer
519
views
The integral of a Gaussian process on a unit sphere
Suppose there exist a zero-mean Gaussian process $\mathbb{G} f_u$ indexed by $u \in \mathcal{S}^{p - 1}$ with known covariance $\mathrm{E} \big[ \mathbb{G} f_u \mathbb{G} f_v \big]$ when both $u$ and $...
- 331
1
vote
1
answer
141
views
Central limit theorem for chi-squared random field on $\mathbb R^p$
Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \...
- 6,853
2
votes
1
answer
155
views
Kalman filter distribution of observation process
Let $(X_t,Y_t)$ be a pair of stochastic processes such that
$$
\begin{aligned}
dX_t =& A_t X_t dt + C_t dW_t,\\
dY_t = & H_t X_t dt + K_tdB_t
\end{aligned}
$$
for some non-random matrix-valued ...
- 5,405
1
vote
1
answer
259
views
Test for OU-Process
Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use?
So far, everything I've seen is hand-...
- 5,405
2
votes
0
answers
109
views
Tightness of Hilbert-space-valued arrays
Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}...
- 83
0
votes
2
answers
251
views
Martingale optional stopping before a stopping time
Here’s an easy one, I hope:
Suppose $\tau$ is a stopping time and $(M_t)$ is a martingale which together satisfy the hypotheses of the optional stopping theorem so that $\mathbb{E}[M_\tau]= \mathbb{E}...
- 3
2
votes
0
answers
173
views
Weak convergence of $\mathcal{L}^2$ valued random variables
Consider two continuous functions $f,g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,\cdot), g(x,\cdot) \in \mathcal{L}^2(\mathbb{R},\mathcal{B},\lambda)$ for all $x \in \mathbb{R}$ and a sequence ...
- 83
1
vote
1
answer
170
views
Stationary distribution of Markov Chain with departure
I have a Markov Chain of $N$ states. Such states represent the energy levels in a molecule.
The states' connectivity is as follows:
States $j\in\{0,\ldots,N\}$ transition to $k\in\{\max(j-M,0),...,\...
- 151
0
votes
1
answer
165
views
About another potential characterization of normal numbers
Normal numbers, in a nutshell, are real numbers that have a "uniform" distribution of digits in standard numeration systems (binary, decimal, and so on.) You can find a formal definition and ...
- 3,098
0
votes
1
answer
1k
views
Convergence in distribution of products
Suppose that a sequence of random variables $Y_n$ convergence in $L^2$ to $Y$, i.e.
$$
E|Y_n-Y|^2\to0\quad \text{as}\quad n\to\infty.
$$
Moreover, there exist constants $c_0$ and $c_1$ such that
$$
0 &...
- 411
2
votes
1
answer
192
views
Upper confidence bound for Poisson process rate parameter
Admittedly, this is an elementary question for mathoverflow. However, I've had no real bites on math and stats.stackexchange so I'm cross-posting.
I am interested in computing an upper confidence ...
- 283
4
votes
1
answer
239
views
Uniform inequality of the form $\text{Proba}(\sup_{v \in [-M,M]^k}|p^Tv-\hat{p}_n^Tv| \le \epsilon_n) \ge 1 - \delta$
Let $M > 0$, $k$ be a positive integer, and $\mathcal V:=[-M,M]^k$. Finally, let $p \in \Delta_k$, (where $\Delta_k$ is the $(k-1)$-dimensional probability simplex) and let $\hat{p}_n$ be an ...
- 6,853
4
votes
1
answer
839
views
A balls into bins problem with combinatorial constraints
We are given $m$ balls and $n$ bins, with $m \ge n$. Each bin can contain at most $c$ balls (we assume that $c$ is an even integer). In a sequential fashion, at each time step, one ball is placed into ...
- 1,677
1
vote
2
answers
302
views
how to derive stationary distribution of maximal entropy random walk
I was reading the paper 0810.4113v2, burda, which analyzed the stationary distribution maximal entropy random walk on the irregular lattice. I am confused on some of the steps.
Description:
The ...
- 211
4
votes
1
answer
423
views
Concentration inequalities on the supremum of average after time $n$
Let $R_1, R_2, \cdots$ be i.i.d. Rademacher random variables (taking values $-1,+1$ w.p. $0.5$). At time $k$, their average is $\frac{1}{k}\sum_{i=1}^k R_i$. One can imagine after $k\geq n$ for some $...
- 183
1
vote
0
answers
61
views
For 1-NN (Next Neighbor), what is the expectation of the largest probability of being the nearest neighbor?
Suppose we sample $n$ points $X_1,X_2,...,X_n$ independently from a distribution $P_X$ on $[0,1]^d$. For a new point $X$ independently from $P_X$, we find its nearest neighbor in $X_1,X_2,...,X_n$. ...
- 11
2
votes
0
answers
169
views
Stochastic Approximation in Reproducing Kernel Hilbert Space
Consider an iterative algorithm with incremental updates
\begin{align}
x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}],
\end{align}
where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
- 1,127
2
votes
1
answer
186
views
Understanding some Hoeffding-type martingale inequality
Would anyone know how to prove the following, coming from the proof of theorem 2 in this paper (https://arxiv.org/pdf/1605.08671.pdf)?
Consider i.i.d. Sub Gaussian random variables $(X_t)_{t\geq 1}$ ...
- 301
0
votes
1
answer
59
views
Looking for a specific kind of a compactly supported one dimensional distribution
I am looking for a sequence of probability distributions (parameterized by $h \in \{1,2,3,4,..\}$) supported on the compact interval $x \sim [a(h),b(h)]$ such that,
$a(h) > \frac{b(h)}{h^{\nu^2}} ...
- 2,246