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.
9,021 questions
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Absolute Continuity of the Karhunen-Loeve expansion coefficients
The Karhunen-Loeve theorem (see these notes or the wikipedia page, for example) states the following:
Theorem: For a continuous, square-integrable, centered stochastic process $(X_t)_{t \in T}$ (with ...
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Measurability of $X$ with respect to $Y$ in conditional probability distributions
Let $\pi$ be a probability measure on $\mathbb{R}^2$ with respective marginals $\mu$ and $\nu$ such that $(X,Y) \sim \pi$.
Notation:
$\pi_{X=x}$ be the conditional distribution of $Y$ given $X=x$,
$\...
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Exchanging the integral and infimum on the space of couplings
Let $\mu,\nu$ be probability measures on $\mathbb{R}^d$ with finite $p$-th moment ($p\in [1,\infty)$) and define the set of couplings by $\mathcal{C}(\mu,\nu)$ i.e. the set of probability measures on ...
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dimensionality reduction of Markov chains
Suppose that $M$ is a time-homogeneous (and, for simplicity, stationary) Markov chain on $d$ states, which induces the probability measure $P$ on paths of length $n$. I seek a Markov chain $M'$ on $d'&...
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Does complete and separable Wasserstein space imply a complete base space?
Also asked on math.SE.
Let $(Z,d)$ be a metric space, and for $p\geq 1$, consider a metric space $(W_p,d_{W_p})$ defined by
The Wasserstein Space $\begin{align}W_p = \{\mu|\mu\textrm{ is a Borel ...
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Is it easier to exit a box to the right of a box in $\mathbb{Z}^2$ if I remove some edges to the left?
Suppose that I am given the graph $G = (V,E)$ where
$V = \{ 1, 2, \dots 2N+1 \} \times \{ 1, 2, \dots 2N+1 \} $
and there is an edge between two vertices $(n,m)$ and $(n',m')$ if and only if
$\vert n-...
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Largest set of $k$-wise linearly independent vectors in $\mathbb F_q^n$?
What is known about the largest set of $k$-wise linearly independent vectors in $\mathbb F_q^n$? I am especially interested when $q=2$, and in the regime where $k$ is fixed an $n\to\infty$. Here are ...
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Distribution of Brownian motion conditional on linear growth
Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.
Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event
$$ E_T := \{|B_s| \geq \lambda s\ \...
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Supremum of sums of functions in $L^1$ taking random signs
Consider the Banach space $X=L^1([0,1])$, and let $n\gg1$ and $x_1, ..., x_n$ be any points in the unit sphere of $X$.
Is there any reasonable lower bound for $$\sup_{(\epsilon_i)_{i=1}^n \in \{-1,+1\}...
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Question/References on the Skorokhod M1 topology
Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
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Generalization of a mind-boggling box-opening puzzle
Motivation. Suppose we are given $6$ boxes, arranged in the following manner:
$$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]$$
Two of these boxes contain a ...
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Higher or lower?
Consider the following game - I draw a number from $[0, 1]$ uniformly, and show it to you. I tell you I am going to draw another $1000$ numbers in sequence, independently and uniformly. Your task is ...
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RMT for modified Wishard matrix $Y'Y$ (where $i$th row of $Y$ is zero if $|x_i^\top u| \le \theta$; else it equals $x_i$)
Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
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Probability of sum of i.i.d. random variables being positive
Let $g,l \in (0,1)$ and $p\in [0,1]$. Let $X(k,1-p)$ be a random variable with binomial distribution with parameters $k$ and $1-p$. Let $Y(k,p)$ be a random variable with binomial distribution with ...
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Show that $\sup_{\|g\|\leq \delta_n}\left| \frac{1}{\sqrt{n}}\sum_{i=1}^n g(Z_i)\right|\rightarrow_{\text{a.s.}}0.$ when $\delta_n\rightarrow 0$?
UPDATE: The result below can be understood as an almost sure stochastic equicontinuity condition. I don't know of any result establishing primitives of almost sure stochastic equicontinuity. If you ...
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Examples of problems in statistics accessible only using information geometry
I am just curious if there are some examples of problems in statistics that are indeed accessible using information geometry while proofs completely avoiding geometry are unknown. In other words, ...
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Oscillation of Riemann-Liouville process after hitting time
Let $W$ be a one-dimensional Brownian motion and let $X_t = \int_0^t (t-s)^{H - 1/2} \mathrm{d} W_t$, $H \in (0, \frac{1}{2})$ be a Riemann-Liouville process. We set
$$ \sigma(a) := \inf \{t > 0 : ...
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Expectation of complex random variable
I am researching frequency offset estimation and ended up reading a paper "Cramer-Rao Lower Bound on Frequency Offset Estimation Error in OFDM Systems With Timing Error Feedback Compensation"...
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Asymptotically small submatrices of random matrices
Consider an ensemble of $N \times N$ random Hermitian matrices distributed according to some unitarily invariant measure
$$P(M) \mathrm{d}M = \frac{1}{Z_{N}} e^{-\mathrm{tr}[ Q(M)]}\mathrm{d}M,$$
for ...
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Sum of RVs satisfying Bernstein condition on moments
Let us say that a RV $X$ with mean $\mu$ and variance $\sigma^2$ satisfies Bernstein condition with a parameter $\beta>0$, if for all $k \ge 2$, it holds that
$$
|\mathbb{E}[(X - \mu)^k]| \le \frac ...
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2
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Can the solution to a controlled SDE with additive noise have non full support?
Let $W$ be a standard $d$-dimensional Brownian motion. Consider the following SDE
$$dX_t = b(X_t, u_t) \, dt + dW_t$$
with initial condition $X_0 = 0$ a.s., $b: \mathbb R^d \times \mathbb R^n \to \...
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Is this process strictly positive?
Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider 1-D stochastic differential equation
$$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$
We assume $\sigma(0) = 0$, and $\...
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Comparison of Rademacher and Gaussian expected values under linear transformations
As per suggestion, I have decided to post the following as a new question, but it is a follow-up to this one: Comparison of Rademacher and Gaussian moments under linear transformations
Let $X$ be an $...
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Echoes of the chord
Just a fun problem I thought of.
A man is playing a magical pipe organ - every chord is an integer number of decibals (dB) loud. The softest chord is $0$ dB. Every chord of $N > 0$ dB creates a ...
- 6,215
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Asymptotic limit of trace of random matrix $(aI_m + WW^\top)^{-1}$, where $W$ has iid rows from $N(0,\Sigma)$
Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a ...
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Does the convergence of $f_n$ imply the convergence of $\mathbb P[\inf_{0\le s\le t}(W_s-f_n(s))\le 0]$?
Let $(f_n)_{n\ge 1}$ be a sequence of non-decreasing and continuous functions defined on $\mathbb R_+$ and taking values in $[0,1]$. Further, for each $t\ge 0$, $n\mapsto f_n(t)$ is non-decreasing. ...
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Hanson-Wright inequality (quadratic form concentration inequality) for bounded random vectors
Is there a concentration inequality for quadratic forms of bounded random vectors $X \in [-1, 1]^n$ with zero mean and given covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ but otherwise ...
CommunityBot
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1
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On the martingale betting scheme
For a fixed probability $0 < p < 1$, let $X^p$ be the martingale that goes up by $1$ with probability $p$, and goes down by $\frac{p}{q}$ with probability $q := 1-p$.
Write $X$ for the ...
- 128k
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Yates-Grundy draw-by-draw sampling inclusion probabilities
Is there an efficient algorithm to calculate the inclusion probabilities (the probability that an item will be included in a sample) in the Yates-Grundy draw-by-draw sampling?
Sampling description: ...
CommunityBot
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Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift
Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
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Renewal Process with inter-arrival times having quadratic tails
Consider a sequence $(X_n)_{n \ge 1}$ of i.i.d. Pareto($2$) random variables, which means
$$
\mathbb{P}( X_1 > x) =
\begin{cases}
1/x \qquad &\text{for } x \ge 1
\\
1 \qquad & \text{else}....
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Coupling/Ordering of Brownian bridges
Suppose I have two 1D Brownian bridges $(B^{(1)}_t,t\in [0,1]),(B^{(2)}_t,t\in [0,1])$, one from $0$ to $0$ and one from $x$ to $y$ where $x,y \geq 0$. Is there a neat way to show that there exists a ...
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1
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For fixed $f \in L^2$ and $T>0$, choose $g$ so that $ \mathbb{E}^x[g(T-\tau)\chi_{X_\tau=1}]=-\mathbb{E}^x[f(X_T)\chi_{\tau \ge T}]$
Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that
\begin{align*}
0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\...
CommunityBot
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A race to the bottom
Nate has a biased coin that comes up heads $\frac{1}{2} + \delta$ proportion of the time, where $0 < \delta \leq \frac{1}{2}$. He is competing against a large number $N$ people who each have fair ...
- 128k
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Analytical approaches to approximate probability density functions of multivariate random functions
Given a random multivariate function $f(x, y, z)$, where $x, y, z$ are independent and identically distributed random variables with a probability distribution $\rho(X)$, I aim to approximate the ...
- 6,696
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Existence of Dirac measures in the context of joint and marginal distributions
Let $\pi$ be the joint law of $(X, Y)$ with marginal distributions $\mu$ and $\nu$. We assume that we have: for all $A \in \mathcal{B}(\mathbb{R})$ such that $\mu(A) > 0$
$$
\nu\left(\{y \in \...
- 63.9k
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Random distance matrices
My question is motivated by the following recent paper:
Gadgil, Siddhartha; Krishnapur, Manjunath, Lipschitz correspondence between metric measure spaces and random distance matrices, Int. Math. Res. ...
- 6,367
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Bialgebras in 1/Kl(D)
$1/Kl(D)$ is the comma category of the one element set in the Kleisli category of the distribution monad. There is mention of it here. The objects are probability distributions called states and the ...
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Smoothness of expectation
Suppose that $X_t$ is a strong solution to the SDE,
$$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...
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Seating assignment inspired question
Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course ...
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Conditional evaluation of random functions
Motivation
Let $(Z(x))_{x\in \mathbb{R}^n}$ be a random function (also known as random field, or random process), i.e. a collection of random variables, but also $Z\in C(\mathbb{R}^n)$ almost surely. ...
CommunityBot
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Strong law of large numbers indexed by a directed set
Let $\xi_{1},\xi_{2},\ldots$ be a sequence of independent random variables with mean 0. For simplicity, assume that each $\xi_{i}$ only takes two values in $[-1,1]$.
Let $\mathscr{F}$ denote the ...
- 12.6k
3
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1
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181
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A nice terminal inequality for martingales
Let $X_t$ be a continuous time martingale taking with $\sup_t \mathbb E[X_t^-] < \infty$, and $X_0 = 0$ almost surely. Assume further that $X_1$ admits a probability density function.
Is it true ...
- 23.2k
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1
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Reverse Doob’s maximal inequality for bounded martingales
Consider the set of discrete or continuous time $L^\infty$-bounded martingales $X$ with $X_0 = 0$ almost surely. Here $L^\infty$-bounded means $\|X\|_{\infty} := \sup_t \mathbb \|X_t\|_{L^\infty(\...
- 23.2k
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Comparing two probability of connection in Bernoulli percolation on $\mathbb{Z}^2$
I want to know for bond Bernoulli percolation on $\mathbb{Z}^2$, does it holds that
$$ \mathbb{P} \left( (0,0)\longleftrightarrow (0,n) \right) \geq \mathbb{P} \left( (0,0)\longleftrightarrow (k,n) \...
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What is a natural interpretation of the commutator of the conditional expectation operator?
Notation: We denote by $\mathbb E_{\mathcal F} X$ the conditional expectation of the random variable $X$ with respect to the $\sigma$-algebra $\mathcal F$.
Given two $\sigma$-algebras $\mathcal G, \...
- 12.9k
3
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1
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Error bound for MonteCarlo estimate of elements in Gram-Matrix
Suppose I have a $n\times n$-symmetric positive-definite matrix $A$ with elements:
\begin{align}
[A]_{ij}=\int_{\Omega}f_i(x)f_j(x) \, dx, \quad i,j=1,\ldots,n
\end{align}
where $\Omega\subset \mathbb{...
- 128k
-1
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1
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Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 6,215
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Sharpening Doob’s upcrossing inequality for Brownian motion
Note: This question is heavily related to a series of posts ([1], [2]) by user GJC20.
Provided a martingale $X$ in continuous-time, Doob's upcrosssing inequality states:
If $U(a,b)$ denotes the number ...
- 6,215
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1
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107
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Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
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