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.
9,358 questions
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Does uniform strict convexity of a local lattice action imply a uniform Brascamp–Lieb inequality?
Consider a sequence of finite-dimensional probability measures $\mu_n$ on $\mathbb{R}^{d_n}$ given by$$\mu_n(dx) = Z_n^{-1} e^{-S_n(x)}\,dx,$$where $x \in \mathbb{R}^{d_n}$, and $Z_n$ is the finite ...
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What proportion of vectors in ${\mathbb{F}_2^n}$ have more than $\frac{n+\sqrt{n}}{2}$ ones
Ben Green's "Finite field models in additive combinatorics" (proof of theorem 9.4) states that for sufficiently large $n$, the set $A$ of vectors in ${\mathbb{F}_2^n}$ with more than $\frac{...
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Expected number of upsets in a knockout tournament
$2^n$ players $P_1, \dots, P_{2^n}$, ordered in decreasing order of skill are placed uniformly at random at the leaves of a binary tree of depth $n$.
They play a knockout tournament according to the ...
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Limit of expectation of a Markov process at stopping times
Suppose $X \in \mathbb R^d$ is a Markov process with cadlag paths and a stationary distribution $\mu$. Let $\tau_n : =\inf\{t\ge 0: |X(t) | \ge n\}$ and assume that $X$ is non-explosive and hence $\...
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Is there a increasing, convex, superlinear $f$ with $c_1 f(x)y \leq f(xy)\leq c_2 f(x)f(y)$ such that $\mathbb{E}[f(X)] < \infty$?
The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference?
Given a positive random ...
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Measurability of $t \mapsto \int_A f(t, \omega)\mathbb{Q}_t(\mathrm{d}\omega)$ when $(t, \omega) \mapsto f(t, \omega)$ is not measurable in $t$
I have a Markov kernel $(t,A) \mapsto \mathbb{Q}_t(A)$ from a standard Borel space $(T, \mathcal{T})$ into another standard Borel space $(\Omega, \mathcal{F})$. Also, for $t \neq s$, $\mathbb{Q}_t \...
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Speed measure for sticky drifted Brownian motion
Consider the following SDE:
$dD_t=1_{\{D_t>0\}}d(B_t+\sqrt{2}t)+dL_t\\dL_t=\sqrt{2}1_{\{D_t=0\}}dt$
under the constrain that $D_t\geq 0$, where $L_t$ is the local time of $D_t$ defined by Tanaka's ...
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Integral representation of Markov operators
On a measurable space $(E,\mathcal E)$, a stochastic kernel is a function $p\colon E\times \mathcal E\to [0,1]$ such that:
for each $x\in E$, the function $A\mapsto p(x,A)$ is a probability measure;
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Cumulants of random inner product on the sphere
Let $Z={U}^\top {V}$ where ${U}$ and ${V}$ are uniformly distributed $\mathbb{S}^{p-1}$. It is known that $Z$ has even moments given by
\begin{align*}
\mathbb{E} Z^{2m} = \frac{(2m-1)!!}{p(p+2)\cdots(...
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solve explicitly an integral equation
Let $\lambda$ be the Lebesgue measure. Let $f \in L_1([0,1])$, I would like to construct a $g$ function in $L_1(\mathbb{R}^+)$ such that
$$
\mathbf{1}_{[0,1]}\lambda(dx)\text{-a.e., }\quad f(x)
= \...
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Martingale central limit theorem: simple version reference
I don't know if it is better to ask here or on MSE, if that's the case I can post the question there. I would need a simple version of the martingale central limit theorem. And, by simple, I mean the ...
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Is the Matérn field $(\kappa^2 - \Delta)^{\alpha/2} u = W$ the stationary distribution of an infinite-dimensional Ornstein-Uhlenbeck SDE?
In spatial statistics, the Matérn Gaussian field on $\mathbb{R}^d$ is often defined as the (weak) solution to the SPDE
$$
(\kappa^2 - \Delta)^{\alpha/2} u \;=\; W,
$$
where $W$ is Gaussian spatial ...
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Inverting the conditional expectation for some coupling
Let $\nu$ be a probability measure equivalent to $\mathbf{1}_{\mathbb{R}_+}(y) \, \lambda(dy)$. Let $\pi$ be a probability measure on $\mathbb{R}^2$ of second marginal $\nu$, such that $\nu(dy)$-a.e.,
...
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Solving decoupleable families of FBSDEs
Let $d_1,d_2\in \mathbb{N}_+$, a stopping time $\tau$, and consider a system of BSDEs
\begin{align}
Y_{\tau}^1 & = \xi^1+\int_{t\wedge \tau}^{\tau}\, f_1(t,Y_t^1,Z_t^1,Y_t^2,Z_t^2)dt - \int_{t\...
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How to prove the convergence of the maximum point random variable of random concave function sequence？
I am really wondering how to prove this lemma from the book 'Counting processes and survival analysis'. No need for the first and second point, just the third point, why does the maximum random ...
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