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
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On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
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expectation of the exponential of the inverse of variable with Marchenko–Pastur distribution
This question is related to another answered before
distribution on the inverse Wishart matrix eigenvalues summation
my question is, is their finite expression for the expectation of
\begin{align}
{\...
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Concentration (or two sided tail bounds around expectations) of maximum and minimum of $n$ iid, subgaussian random variables
I asked this on MSE, but got no answer, hence asking here now. Help appreciated!
My question is motivated by this question and this question, where the first was aimed for giving a one sided tail ...
- 1,467
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"Probability" for a partitioned matrix to be singular
Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix
$$
M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
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What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$
Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
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Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$
I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that
If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...
user157103
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Express inclusion-exclusion principle in terms of matrix operations
First of all i denote $\{1,2,3,...,m\}$ by $[m]$
Let there be a collection of sets $\alpha=\{A_{1},A_{2},...,A_{m}\}$ such $\bigcup_{i\in[m]}A_{i}\subseteq [n]$
Consider any function $f:\mathcal{P}([...
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Finding a square integrable dominating function for function class
problem statement
For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$
where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...
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Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix
Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
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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}...
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A distribution of maximum of sums if add to the minimal
Consider a vector of $n$ integer variables with initial values of 0. Each step we take random $w_i\thicksim NB(q, l)$ (independent randon values with the same negative binomial distribution) and add ...
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Lower bound on probability of sum of independent random variables
I'm interested in estimating the following quantity
\begin{equation}\lim_{k \to \infty} \sum \limits_{i=1}^l \sum \limits_{l_1 = 0}^{i} {{nk-k}\choose{l_1}} \Big( \frac{1}{k}\Big)^{l_1} \Big( \frac{...
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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 ...
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Hypothesis testing for not identically distributed random variables conditioned on the outcome of a subset
I encountered the following problem (I give more details of the problem at the end of the post) and I am trying to figure out the best way of performing a null hypothesis testing. I looked for similar ...
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Is there any relation between moments of random matrix and its eigenvalue distribution?
Let $\mathbf{X}$ be a random matrix with independent Gaussian random variable entries with different variances $v_{ij}$. Also define $\mathbf{A}=\mathbf{X}^\mathrm{H}\mathbf{X}$. Is there any relation ...
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Deriving asymptotic variance of generalized estimating equation estimator (GEE)
As well known to us, K.Y. Liang and S. Zeger proposed GEE for longitudinal data analysis in their famous paper[1]. At the appendix of the paper, authors show the proof of Theorem 2. I tried to ...
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Construct a random time such that the strong Markov property of Brownian motion fails
Let $\{B_t, \mathcal{F}_t; t\ge 0\}$ be a standard, one-dimensional Brownian motion. Can we construct a random time $S$ such that $P[0\le S < \infty] = 1$ and $W_t = B_{S+t} - B_S$ is not a ...
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Joint typicality of sequences
I know that for two i.i.d distributions $P$ and $Q$ the probability that $Q$ will produce a length $n$ sequence that is $\epsilon$-typical according to $P$ is bounded by
$$Q(T_{P,\epsilon})\leq e^{-nD(...
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Tightness of a uniformly bounded sequence of functions integrated with respect to a semimartingale
I am reading this paper by Jacod, Jakubowski and Mémin. In the proof of Theorem 1.3 the authors define, for each $n\geq1$ the function $\phi_n$ by
$\quad\phi_n(s)=i+1-ns,\quad\text{if } \frac{i}{n}&...
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hypergraph product that preserve expansion properties
I am looking for a hypergraphs product of hypergraph H1,H2 that preserves some expansion properties of H1,H2.
The expansion property I am looking at is HD-random walk.
The product I am looking for is ...
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Distance between two sample quantiles
Let $X_1,\dots X_n$ be i.i.d. samples from an unknown distribution. We know the distribution has uniformly bounded probability density function $f(x)$. Let $1>\tau_1>\tau_2>0$ be two quantile ...
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If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
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Functions that preserve the mixing of a stochastic process
Suppose we have a continuous-time stochastic process $s(t)$ that's mixing. What properties would a function $f$ have to have so that $f(s(t))$ would also be mixing?
I'm sure this is a well-known ...
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Bounds on $\inf_{x,x' \in \mathbb B_X}TV(P+x,Q+x')$, where $P$ and $Q$ are distributions with density on the space $X=(\mathbb R^n,\ell_p)$
Let $n \ge 1$ be an integer, $p \in [1,\infty]$, and $P$ and $Q$ be two (probability) measures on the metric space space $X=(\mathbb R^n,\ell_p)$ which have densities w.r.t the Lebesgue measure on $X$,...
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Uniformly bounded sequence {X_i} of independent random variables series converges or diverges depending on bejavior of variance?
Let $\{X_i\}$ be a uniformly bounded sequence of independent random variables. Does $\sum_{i=1}^{\infty}X_i-E[X_i]$ converges or diverges, depending on whether $\sum_{i=1}^{\infty}\sigma_i^{2}$ ...
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Hitting order of sets by a Lévy process
Let $X$ be a transient Lévy process on $\mathbb R$, and $B\subseteq \mathbb R$ a Borel set with first hitting time $T_B = \inf \left\{t>0 : X_t\in B\right\}$. For Borel $A\subseteq B$, can anything ...
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Reference request: mathematical expectation of a random object in a topological space
Recently I got interested in the following question: what does a mathematical expectation look like for a random object taking values in a topological space?
This turns out to be a difficult ...
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Malliavin derivative of Ito process
Let $X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$ where $\mu$ and $\sigma$ are $C^1$ functions satisfying the usual growth restriction and $W_t$ is a $d$-dimensional Brownian motion. ...
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Bayesian Bandits - What's the probability that choice K is the best?
I have $K$ very unfair coins. I don't know how unfair they are, but they all seem to have different probabilities of landing heads. I'd like to figure out which one is best as quickly as possible.
...
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Ask for some percolation reference textbook
I try to learn Bernoulli percolation recently. Could anyone provide some lecture notes or textbooks to enter this field? Thanks.
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A problem with the dual form of semi-discrete optimal transport
Consider the uniform distribution $\lambda$ on $[0,1]$, and a point measure $\rho$ with density $\frac{1}{2} (\delta_{x_1} + \delta_{x_2})$, where we have $0\le x_1 \le x_2 < 1/2$.
If our cost is ...
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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: ...
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Specific property of borelian sigma-algebras
Let X be a set and S a sigma-algebra on X.
Let us name borelian sigma-algebra on X a sigma-algebra that is generated by a topology T on X. Given that it is possible for a set X that some sigma-...
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Wiener measure in the space of functions of two or more variables
Wiener measure in the space of continuous functions of two variables had been introduced by J. Yeh in the 1960 paper "Wiener Measure in a Space of Functions of Two Variables" (AMS free access)
(p. ...
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A question about positive operator pregenerator [closed]
Thank you for reading.
My question was raised up when I tried to prove an example in the book of Liggett(1985), which is in P13 Example 2.3(a).
Here is a link of the page:
https://books.google.com/...
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Eigenvalue density for correlated samples
Suppose we generate $P$ samples $\mathbf{x}^1, \dots, \mathbf{x}^P$ from a multivariate normal distribution of dimension $N$, mean zero and covariance $\Sigma$.
Let $\mathbf{C}=\frac{1}{P}\sum_{k=1}^...
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About martingales induced by iterative processes
Suppose I have a discrete stochastic process $\{ X_i \}_{i=1,\ldots..}$ defined as, $X_{i+1} = X_i - \eta \nabla f(X_i) + \sqrt{\eta} \xi_i$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and $\xi_i \...
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Skorohod Space with $J_1$ topology homeomorphic to Frechet Space
Is the Skorohod space $D([0,T];\mathbb{R}^d)$ equipped with the $J_1$ topology homoeomorphic to a separable Fr\'{e}chet space. In particular, is it homeomorphic to $L_{\mu}^1(\mathcal{B}([0,1])$ ...
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Concentration of Sample Mode
Is there a concentration bound for sample mode, when there exists a unique mode for a density $f$ that doesn't depend on $f$ (Essentially I am looking for a Chernoff type bound for mode)?.
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Optimizer of a semi-discrete optimal transport problem
Provided two probability distributions $\mu(dx)=\rho(x)dx$ and $\nu(dx)=\sum_{i=1}^n p_i\delta_{y_i}(dx)$ that are supported on some measurable set $\Omega\subset\mathbb R^d$, we consider the semi-...
user128095
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Computing the partition function via one of the three methods
I am trying to compute the averaged partition function for some system (with very large $N$) and I reach this point:
\begin{equation}
\left < Z\right > =\int \prod_i^N \left (\frac{\...
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Role of Brownian Filtration
I'm trying to understand papers around Novikov, Kazamaki, Kramkov, Shiryaev...
and there is something I can't figure out. I will try to describe it shortly and can give more information, if needed.
...
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Girsanov Theorem - Dependence of the probability space
I'm trying to understand the proof Novikov Condition and other works in this field (Kazamaki etc) and for this, i have to understand the Girsanov Theorem, which is also a part of the proof. In ...
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A closed form of mean-field equations
Assume that a system at time t, for example number of costumers in a line at time $t$ which is denoted by $q(t)$, follows a Markov chain with these dynamics (probabilities)
$$P(q(t+\Delta t)-q(t)=1)=\...
user135936
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What is the probability that $X_i$ is the $k^{th}$ order statistic in consecutive trials?
Consider $n$ r.vs ${X_1, X_2,...,X_n}$. Each is i.i.d drawn from some distribution $f(.)$. What is the probability that $X_i$ is the $k^{th}$ order statistic in any two consecutive trials?
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Knights on an n x n chessboard
In a n x n chessboard a white knight sits on the top left corner, and a black knight on the bottom right corner. Starting with white, the two knights take turns to move at random, and with equal ...
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Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
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Asymptotic upper densities in infinite binary stochastic processes
Consider an infinite binary process $X=X_1,X_2,\ldots$ (with corresponding probability $P$). For some bits $1$ is less probable than $0$. I am interested in the following asymptotic upper density :
$$\...
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Probability of detecting small bias in a die in the low confidence regime
We are given a biased $m$-sided die: one of the sides has probability $\frac{1}{m} + \gamma$ and all the rest have probability $\frac{1}{m} - \frac{\gamma}{m-1}$ each. The goal is to figure out which ...
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Taylor series expansion of quantile function
Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $.
We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$.
Do you have any ...
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