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.
9,022 questions
0
votes
1
answer
2k
views
Stationary distribution in general Markov Chains
This is just a reference request for a result which is very general, useful and should be well-known, but I've failed to find a good reference to cite.
The problem is to define the "most natural" ...
- 1,341
0
votes
1
answer
369
views
How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X
Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...
0
votes
1
answer
599
views
A basic question on necessary and sufficient condition for positive recurrence
If state $j$ is recurrent and the following holds can it be called as positive recurrent ?
$$\lim_{n -> \infty}\frac{1}{n}\sum_{k=1}^{n}p_{jj}^{(k)} > 0$$
I know that this a necessary ...
- 209
0
votes
1
answer
560
views
Random walk on the hypercube
Consider the hypercube $Q_4$. I would like to know how to compute the number of steps of a random walk in this graph such that the probability to be at a vertex is a given number $x$. I think I just ...
- 119
0
votes
1
answer
502
views
Mathematical properties of financial prices
Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes.
What is known about their mathematical properties ?
I know ...
- 24.9k
0
votes
1
answer
61
views
What is the distribution of the distance between a specific word in a Text which is generated by a markov process?
What is the distribution of the distance between a specific word in a Text which is generated by a markov process?
For example for a text which is generated by a multinomial distribution over words, ...
- 87
0
votes
1
answer
407
views
Transformation of probability space.
Let (\Omega, F, P) be a probability space, which may have atoms (important), S be a set of measure-preserving transformations T:\Omega\to\Omega, that is, such that preimage T^{-1}(A) is measurable ...
- 1
0
votes
2
answers
763
views
multivariate distributions unaffected by unitary transformations
Hi,
In my research I reached some very nice results for IID complex Gaussian vectors $\bf{x}$.
Now I realize that my results hold for any random vectors that are unaffected by a unitary map, i.e., $\...
- 1
0
votes
2
answers
571
views
Number of transitions of a markov chain in a time interval
Let us consider the homogeneous continuous time Markov chain $(X_t)_{t\ge 0}$ with two states {0,1} and the intensity matrix
$Q=\begin{pmatrix}-\lambda& \lambda\\\ \mu& -\mu\end{pmatrix}$
...
0
votes
2
answers
257
views
Efficient computation of $E\left[\frac{1}{1+\sum_iX_i}\right]$ where $X_i$ is RV with Bernoulli distribution with different probabilities
Suppose we have the random variables $X_1, \ldots, X_n$ that have Bernoulli distributions with the (possibly different) probabilities $p_1, \ldots, p_n$. For example, $X_1$ = 1 with probability $p_1$ ...
- 21
0
votes
2
answers
722
views
Concentration bound using Azuma's inequality and Law of total probability
Given a function $f(X_1,\cdots,X_n,Y)$ on random variables $\{X_i\}$ and $Y$, which is continuous ,
I want to
show that $f$ concentrates around its expectation $\operatorname*{E}[f]$, i.e., a formula ...
0
votes
2
answers
449
views
Chances of streaks in small bit-streams
Let's say a series of 10 bits is output randomly. Now lets do that 256 times. I'd like to find out what the expected number of streaks of 1s or 0s are for each of the possible sizes 1-10.
For example,...
- 123
0
votes
1
answer
485
views
Estimating probability of set membership
I have a number of discrete finite sets, $A_0$ through $A_n$. I do not actually know their contents, but I know the size of each set and the size of the intersection between $A_0$ and each of the ...
0
votes
2
answers
60
views
Do continuous martingales satisfy this nice terminal inequality?
Let $X$ be a continuous, non negative martingale on $[0, 1]$ with $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. Assume further that $X_1$ admits a probability density function. Is it true that the ...
- 6,223
0
votes
1
answer
160
views
Dot product of a randomly orientated vector and a fixed vector
Let us consider a random variable $Z$ with a probability density function $f$ with respect to the Haar measure on $\mathrm{SO}(3)$. Next, we consider two fixed normal vectors $u,v$ in $\mathbb{R}^3$. ...
user528399
0
votes
1
answer
100
views
Projection on a countable union of linear subspace
For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is ...
- 31
0
votes
1
answer
150
views
Property of $p$-norm in the $n$-simplex
Let $\mathbb{S}^{n}$ be the canonical simplex of $\mathbb{R}^{n}$ and let $u = (1/n,\dotsc,1/n)$. Is it true that
$$\lVert x - u \rVert_p \leq \lVert y - u \rVert_p$$
implies that
$$\lVert x\rVert_p \...
0
votes
1
answer
207
views
Some continuity issues of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
0
votes
1
answer
184
views
What conditions should be satisfied for a rational function to be a moment generating function?
I have a table of points at which a moment generating function is evaluated (for points $t_0,t_1,t_2,\ldots,t_n$ I know $M(t_0), M(t_1), M(t_2),\ldots,M(t_n)$).
I've approximated these tabular ...
- 49
0
votes
1
answer
69
views
Correlation for a Sum of random vectors from the sphere multiplied by matrices
Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be some matrices. Suppose we sample $x_1,\dots,x_n,y\sim \mathcal{U}(\mathbb{S}^{d-1})$, where $\mathcal{U}(\mathbb{S}^{d-1})$ is the uniform distribution ...
- 155
0
votes
1
answer
231
views
Concentration inequalities for random sampling without replacement
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
- 105
0
votes
1
answer
54
views
How is this interpolating curve well-defined in the minimizing movement scheme?
Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
- 825
0
votes
1
answer
88
views
Can we lower bound this entropy by $\int_{\mathbb R^d} \rho^k (x) \, \mathrm d x$ and $\int_{\mathbb R^d} |x|^2\rho (x) \, \mathrm d x$?
We define $U : [0, \infty) \to [0, \infty)$ by $U(0) := 1$ and $U (s) := s \log s + (1-s)$ for $s >0$. Then $U$ is strictly convex. The minimum of $U$ is $0$ and is attained at $s=1$. Let $\mathcal ...
- 825
0
votes
1
answer
154
views
Non-negativity of stochastic integral with indicator, Meyer-Tanaka Local Time
Consider the following stochastic integral:
$$
X_t := \int_0^t \mathbb{I}_{ \{ W_s \geq 0 \}}\, dW_s.
$$
Is $X_t$ almost-surely non-negative?
Using this answer, it seems that
$$
X_t = \max( W_t, 0) - \...
- 109
0
votes
2
answers
280
views
Bounds tighter than the additive Chernoff
Additive Chernoff
Suppose $X_1, \ldots, X_n$ are i.i.d. random variables, taking values in $\{0,1\}$. Let $p=\mathrm{E}\left[X_i\right]$ and $\varepsilon>0$.
\begin{gather*}
\operatorname{Pr}\left(\...
- 105
0
votes
1
answer
103
views
Probabilistic bounds of random polynomials
This is follow-up question to my previous question about the expected number of roots .
I am considering a random polynomial given by $$p(z) = \sum_{i=0}^{n} a_i z^i$$,
where each coefficient } $a_i$ ...
- 826
0
votes
1
answer
165
views
Does this inequality hold for the cumulant generating function?
Suppose a random variable $X$ is zero-mean and the cumulant generating function is
$$
K\left( t \right) =\log \mathbb{E}[e^{tX}].
$$
Given any positive constant $\tau > 0$, does this inequality
$$
\...
- 49
0
votes
2
answers
239
views
Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices
I am working with two random matrices, $Z$ and $H$:
$Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$.
$H$ is a $K \times K$ ...
- 37
0
votes
1
answer
87
views
Is the $2$-point function translation invariant for general Gaussian meaures?
Let us consider the real Hilbert space $H:=L^2\bigl(\mathbb{R}^n, \mathbb{R}^n\bigr)$ and "any" centered Gaussian measure $d\mu$ on it.
Next, denote a generic element of $H$ by the column ...
- 3,477
0
votes
1
answer
154
views
Joint distribution of randomly permuted Poisson random variables
Let $U_1, ..., U_n$ be Poisson random variables with rates $ \lambda_1, ..., \lambda_n$ such that $\lambda =\sum_i \lambda_i = O(1)$ (i.e the sum of the rates is bounded). Suppose we have $n$ buckets. ...
- 662
0
votes
1
answer
112
views
Where does this coupling result use independence when bounding total variational distance?
I am reading this paper, which gives the following coupling result:
Throughout this, I'll assume the dimension $k$ is clear. Let $e_i$ be the $i$-th basis in the $k$ dimensional standard basis.
A $k$ ...
- 662
0
votes
1
answer
67
views
Multivariate random variable problem [closed]
I'm stuck on this problem:
Let a random vector $X$ be given in $\mathbb{R^{10}}$ with the standard scalar product. It is known that $\mathbb{E}[XX^T] = 5I_{10}$, $I_{10}$ – identity matrix of order 10....
- 69
0
votes
1
answer
124
views
Definition of sequence sampled from a measure
Question: Exactly what does it mean for a sequence of points to be sampled from a given probability measure?
I have in mind statements such as «let the sequence $(x_k)$ be sampled with density $f$», ...
- 21
0
votes
1
answer
940
views
Derivative of log-likelihood function for Gaussian distribution with parameterized variance
Suppose we have a parameter $\theta \in R^{n}$ that defines some noisy observation $z=\mu(\theta)+\eta, z\in R^{m}$ where the noise follows a Gaussian distribution whose covariance is a function of ...
- 75
0
votes
1
answer
292
views
Proofs on the convergence of optimization algorithms
I was reading the following link (https://en.wikipedia.org/wiki/Scoring_algorithm) on the "Fisher Scoring Algorithm". As I understand, the Fisher Scoring Algorithm is similar to the Newton-...
- 11
0
votes
1
answer
182
views
Deducing norm concentration from MGF bounds
Suppose that $X$ is a centered, $\mathbf{R}^d$-valued random variable such that for all $t \in \mathbf{R}^d$, there holds the bound $$\log \mathbf{E} \left[ \exp \langle t, X \rangle \right] \leqslant ...
- 801
0
votes
1
answer
51
views
Convergence of Gaussian measures $\{ d\mu_a \}$ whose variances depend smoothly on the index $a$
Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function such that $f(x)$ is positive in a small punctured neighborhood of $x=0$ but $f(0)=0$.
Now, define a collection of centered Gaussian measures on $...
- 3,477
0
votes
1
answer
110
views
Positivity of linear combination of gaussian variables
Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
- 49
0
votes
1
answer
350
views
The expected value of a double exponential function of normal random variable
Let $X$ be a random variable from a normal distribution $N(\mu, \sigma)$. How do we calculate the expectation $E[e^{k\cdot e^{-X}}]$, where $k<0$?
I think we can use the moment generating function
...
- 13
0
votes
1
answer
201
views
Infinite limit of sums of gamma functions is constant?
The following expression arises in the study of hierarchical models. I suspect that the sum of the underlined $4$ terms become constant as $\alpha\rightarrow \infty$. Mathematica agrees when prompted ...
- 111
0
votes
1
answer
91
views
Can a measure on a finite metric space be Alhfors regular?
Recall that a probability $\mu$ measure on a metric space $(X,d)$ is called Ahlfors $q$-regular if there are $0<c\le C$ such that: for $\mu$-a.e.\ $x\in X$ one has
$$
cr^q \le \mu(B(x,r)) \le Cr^q,
...
- 364
0
votes
1
answer
139
views
Lévy measure and jump behaviour of the corresponding Lévy process
Let $(X_t)_{t \ge 0}$ be a Lévy process on $\mathbb R$ with Lévy measure $\nu$.
Define the jump counting measure $N(t, A) = \lvert\{s \in [0, t] \mathrel: \Delta X_s \in A\}\rvert$
where $\Delta X_s$ ...
- 77
0
votes
1
answer
218
views
Is the unconditional variance of a RV an upper bound for the variance of any conditional expectation of the RV?
Let $X$ and $Y$ be continuous random variables with finite first and second moments. Then, is it true that $Var[X]\geq Var[E(X|Y)]$?
0
votes
1
answer
115
views
Reference request: Gaussian branching processes
(Q1) Are there known constructions of branching general Gaussian processes (preferably in continuous time)? Something like branching fractional Brownian motion or OU.
Also, (Q2) what are the modern ...
- 620
0
votes
1
answer
77
views
Meyer's example of a separable process with no path regularity
This question is a cross-post from math.stackexchange.com. I am reposting it here since I didn't receive an answer there. The original post can be found by this link.
In the following excerpt from ...
- 620
0
votes
1
answer
159
views
Approximation of a random sum of random variables (infinitely divisible distribution) by a triangular array
We know that a Poisson distribution can be approximated by a binomial distribution. More exactly, let $(X_{jn})_{1\leq j \leq n}$ be a i.i.d. triangular array such that
$$P[X_{jn}= 1 ] = p_n = 1- P[X_{...
- 135
0
votes
1
answer
84
views
Can we find the following $k$ so that the following inequality holds for asymptotic normal?
Following this question:Can we find such $k$ so that the following inequality holds?.
Consider a sequence of independent $n-$dimensional random vectors $u, v_1, v_2,\dots, v_k$ uniformly distributed ...
- 288
0
votes
1
answer
149
views
Second moment of stochastic integral wrt Levy Processes
I have a question about the second moment of the integral wrt Levy Processes.
Let Z a Levy processe. We know that:
And a few page later is written that by differentiation of the characteristic ...
- 77
0
votes
1
answer
272
views
Change of measure formula for the Föllmer process
While reading a preprint Eldan, Lehec, and Shenfeld - Stability of the logarithmic Sobolev inequality via the Föllmer Process I came across the following SDE in Section 3:
$$d X_t=d B_t+\nabla \log P_{...
- 537
0
votes
2
answers
182
views
Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)
Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
\mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...
- 135