All Questions
Tagged with pr.probability st.statistics
317 questions with no upvoted or accepted answers
2
votes
0
answers
96
views
Limiting value of $\dfrac{1}{m}\mathrm{tr}(FAF^\top (FBF^\top)^{-1})$, where $F$ has iide $N(0,1)$ entries and $A,B$ are deterministic
Let $F=F_{m,d}$ be a random $m \times d$ matrix with iid entries from $N(0,1)$. Let $A=A_d$ and $B=B_d$ be deterministic $d \times d$ positive-definite matrices. In case it helps, it may be assumed ...
- 6,853
2
votes
0
answers
87
views
A complex problem involving densities (likelihood functions) and optimization
Consider the following autoregressive process with normal errors:
\begin{equation}\label{7YlUV4i8nuO}\tag{I}
y_t = \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim} N(0,\sigma^2)
\end{equation}
We ...
- 13
2
votes
0
answers
62
views
Probability bounds of some ranked version of Dirichlet distribution
Recently I have come across a distribution defined on the open ranked simplex $\nabla^{n-1}_+ = \{\vec x \in \mathbb{R}^n:\sum_{k=1}^n x_k =1, x_1 \geq x_2 \geq \cdots \geq x_n > 0\}$, whose ...
- 21
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
2
votes
0
answers
138
views
Optimal Monte Carlo Trace Estimator
For a psd real symmetric $d\times d$ matrix $A$ and a function $f: \mathbb{R}^d \to \mathbb{R}$, with $f(x) := x^T A x$ we have that with $p(x) = \mathcal{N}(0_d, I_d)$ (i.e. standard multivariate ...
2
votes
0
answers
61
views
Approximate logarithmic bound on expected maximum via central limit theorem
If $Z_i$ are standard normal, possibly dependent, one can show that
$$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$
I'm looking for a similar (asymptotic) bound for asymptotically normal ...
- 203
2
votes
0
answers
122
views
Consistent approximation of weighted Radon transform of smooth probability density, using kernel density estimation
Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, ...
- 6,853
2
votes
0
answers
87
views
The covariance of certain random variable
We define two random variables $X_n,Y_n $ on the sample space $\{1,2,3,\cdots,n\}$ with counting measure. We denote by $C_n$ the covariance of theses two random variables: $C_n=Cov(X_n,Y_n)$.
...
- 356
2
votes
0
answers
51
views
Spectral approximation of $(XX^\top/d)\circ(X\Sigma_dX^\top/d)$ where $X$ is an $n \times d$ random matrix with iid rows from $N(0,\Sigma_d)$
Let $X \in \mathbb R^{n \times d}$ be a random matrix with iid rows from $N(0,\Sigma_d)$ where $\Sigma_d$ is a $d \times d$ psd matrix verifying w.h.p,
$\mbox{trace}(\Sigma_d/d)= 1$.
$\|\Sigma_d\|_{...
- 6,853
2
votes
0
answers
172
views
Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix
Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...
- 6,853
2
votes
1
answer
415
views
High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)
Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
- 6,853
2
votes
0
answers
140
views
Adding elements in a list *in expectation*
Suppose $𝐿_1,…,𝐿_𝑘$ are lists with $n$ elements each. We use a fully independent hash function ℎ to compute a value for each element of each list. (We suppose the hash function returns a value ...
- 43
2
votes
0
answers
49
views
What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?
I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any.
I'm looking at the description of a short-term position in ...
- 223
2
votes
0
answers
68
views
Approximate any point of the interval $[-1/2,1/2]$ by the sum of $n$ iid uniform random variables from $[-1,1]$
Let $x \in [-1/2,1/2]$ and $X_1,\ldots,X_n$ be drawn iid from the uniform distribution on $[-1,1]$.
Question. Given $\varepsilon \ge 0$ an integer $k \in [1,n]$, what is a good lower-bound on the ...
- 6,853
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
2
votes
0
answers
2k
views
Approximate Multivariate Gaussian Integration by Parts
When $Z$ is a $\mathcal{N}(0,1)$ random variable, $f$ smooth from $\mathbb{R} \to \mathbb{R}$ we have the Gaussian integration by parts formula
$$
\mathbb{E}(Zf(Z)) = \mathbb{E} f'(Z).
$$
One analog ...
- 21
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
2
votes
0
answers
244
views
An inequality regarding centered Bernoulli random variables
Let $\left\{V_1,\ldots,V_n\right\}$ be a set of (possibly dependent) identically distributed Bernoulli random variables, with
$$ p = \mathbb{P}\left(V_i=1\right) = 1-\mathbb{P}\left(V_i=0\right),\quad ...
- 159
2
votes
0
answers
80
views
Bridging between Rosethal Inequalities and log convex tails
Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $\|X\|_p = (E|X|^p)^{1/p}$.
Then we have the classical "Rosenthal-type ...
2
votes
0
answers
107
views
Expectation of a Random Matrix that Contains Wishart Form
I am interested in calculating the expectation of the following random matrix:
$$A=WX(X^TWX)^{-1},$$
where $W \sim W_p(n,I)$ is a $p\times p$ random Wishart matrix, and $X$ is a fixed $p\times m$ ...
- 43
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
0
answers
46
views
increasing inter-class distances results in decreasing linear regression error
Let $\{\mathbf{x}_i, y_i \}$ be a set of binary-labeled samples ($\mathbf{x}_i \in \mathbb{R}^d, y_i \in \{a,b\}, a,b\in\mathbb{R}$). Let $\{ \mathbf{x}'_i, y_i \}$ be also such a set.
Define $\mathbf{...
- 183
2
votes
0
answers
149
views
Min Max Equality in Information Theory
Let $\mathcal{Y}$ and $\mathcal{X}$ be finite sets and let $Q_Y$ be a fixed probability mass function on $\mathcal{Y}$. Also, let $P_{X | Y}$ be some fixed conditional distribution on $\mathcal{X} \...
- 121
2
votes
0
answers
448
views
Relation between pseudo-dimension and Rademacher complexity
With techniques of Dudley's entropy bound and Haussler's upper bound one can show that there exists a constant $C$ such that any class of $\{0,1\}$ indicator functions with Vapnik-Chervonenkis ...
- 517
2
votes
0
answers
171
views
Distribution for the extreme values of a cumulative sum of normal variables
Suppose I have a sample $X$ of $n$ iid Normal random variables $(X_1,X_2,..,X_n)$.
Now, define the sample mean $u=\frac1n\sum_{i=1}^n X_i$ and let $Y_i=X_i-u$.
Let $Z_k=\sum_{i=1}^k {Y_i}$. Note ...
- 121
2
votes
0
answers
74
views
Best estimator for a 3 coin problem
Let $X,Y,Z$ be three unfair coins. We consider the coin $\Gamma=XI(Z=1)+YI(Z=0).$ We are given the sample $S=\{(\Gamma_i,Z_i)| i \leq n \}$ Let $k=\sum_{i \leq n}I(Z_i=1)$. For every pair of $(\...
- 103
2
votes
0
answers
101
views
Best describing a stochastic process in terms of others
Intuitive Question
Suppose I'm given a set of $k$ time-series $\{X_t^1,\dots X_t^k\}$. Is there a way to determine how much of each series is dependent on the others.
Formal Question
More ...
- 5,405
2
votes
0
answers
80
views
Maximal and second max of chi-squared (normal squared) distribution
Suppose $X_i$ are i.i.d.~$\log \chi^2$ where $\chi\sim N(0,1)$ distribution, in which case
$$
F(x) = \mathbb{P}\left( \log \chi^2 \le x \right)
= 2\Phi(e^{x/2}) - 1,
$$
and
$$
f(x) = e^{x/2}\phi(e^{x/...
- 719
2
votes
0
answers
54
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
2
votes
0
answers
87
views
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 (...
- 2,246
2
votes
0
answers
619
views
Laplace transform of a integral function of CIR/CEV process
The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
- 323
2
votes
0
answers
366
views
Convergence rate of Pearson correlation matrix
I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
- 121
2
votes
0
answers
60
views
Consistency of M-estimators when the constraint set also has to be estimated
Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions.
Assume we have the following convex optimization problem:
$$
\...
- 123
2
votes
0
answers
386
views
What is the concentration of measure for Gaussian random variables which are independent, but are transformed?
This might be a too easy question for Mathoverflow, but Googling led to similar questions and answers here (though not the one I was looking for).
The question is split into two:
I have a matrix $X \...
- 131
2
votes
0
answers
175
views
Implication of MGF inequality
Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs.
It is known that X and Y have the same CDF iff they have the same MGF.
My ...
- 517
2
votes
0
answers
1k
views
Converse for Levy's continuity theorem
Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)...
- 907
2
votes
0
answers
98
views
Finding a general form of the density function when we have a four dimensional random variable
Consider a subject having time of the specific event $T_i$, which is a single sample from a
distribution $F_i$ with density $f_i$ and support
$[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
- 35
2
votes
0
answers
71
views
Asymptotic results for functions of order statistics
There are $n$ ($n \ge 3$) iid random variables $\{ {c_i}\} _{i = 1}^n$ on the interval $[\underline c,\bar c]$ ($\underline c>0$). The cdf $F(\cdot)$ and pdf $f(\cdot)$ are unkown to us, but we ...
- 29
2
votes
0
answers
191
views
MLRP of random variables and order statistics
Suppose we have $N$ independent random variables $X_1, \cdots, X_N$ drawn from $f_1 > \cdots > f_N$ where $f_i > f_j$ indicates that $f_i$ and $f_j$ satisfy the monotone likelihood ratio ...
- 21
2
votes
0
answers
341
views
Marginalizing multivariate normal over defined interval
Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} \;...
- 21
2
votes
0
answers
687
views
Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable
I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...
- 21
2
votes
0
answers
265
views
Expectation of a multivariate Gaussian over a plane
For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :
$E[X|X^Tb = c]$
...
- 21
2
votes
0
answers
90
views
Limiting distribution of the cardinal of a Markovian set
Let $S_1=\lbrace u_1 \rbrace$ where $u_1$ is a random uniform drawing on $[0,1]$. To build $S_{n+1}$ draw $u_{n+1}$ uniformly on $[0,1]$ (independently from previous draws) and draw $v_{n+1}$ ...
2
votes
0
answers
271
views
Convergence of sample mean
I have a two-index succession of real-valued random variables $x_{t,n}$ such that $\lim_{n\to\infty} x_{t,n} = x_t$, for all $t$ and suitable limit r.v. $x_t$.
I would like to prove that $$\lim_{n\to\...
- 20.2k
2
votes
0
answers
1k
views
Random variables: multivariate second-order Taylor approximation (delta method)
Let $g:\mathbb{R}^2\rightarrow \mathbb{R}$ be a smooth, but not necessarily bounded function and $X$ and $Y$ two random variables that are not independent. (assuming they yield sufficiently many ...
- 21
2
votes
0
answers
422
views
Generalizations of Gram-Charlier and Edgeworth series?
I am looking for references pertaining to, and/or help in deriving, generalizations of the Gram-Charlier and Edgeworth series for non-Gaussian reference probability distributions.
I would like to ...
- 1,890
2
votes
0
answers
979
views
How to calculate/approximate expectation of function of a binomial random variable?
Hi,
I am stuck at following problem in my research.
Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. $\...
- 21
2
votes
0
answers
1k
views
Moments of function of Poisson process
(I'm new to Poisson processes, so please edit if my terminology is incorrect.)
Edit: per comments, here is a (more) general version of the originally posted problem (which is now at the bottom, below ...
- 101
2
votes
0
answers
530
views
About generalization of stirling numbers of the second kind
Hello,
The Stirling numbers of the second kind count how many ways can a set of $k$ elements be partitioned into $n$ non-empty classes, with $k=n,n+1,\dots$.
My question is: Is there a ...
2
votes
0
answers
548
views
What will be the distribution of harmonic mean of two correlated gamma random variables?
Suppose there are two correlated random variables $X_1$ and $X_2$ both are gamma distributed but having different shape and scale parameters with correlation coefficient $\rho$. What will be the ...
- 133