Skip to main content

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.

41 questions from the last 30 days
Filter by
Sorted by
Tagged with
-2 votes
0 answers
7 views

Convergence of measures in the Lévy–Prokhorov metric and weak convergence of measures

How to prove that over R the convergence of measures in the Levi-Prokhorov metric is equivalent to the weak convergence of measures
S4SKE's user avatar
  • 1
1 vote
0 answers
42 views

Measurability of a map involving probability measures

Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
triple_sec's user avatar
2 votes
0 answers
32 views

Order of $\mathbb{E}[ \max_i |x_i + z_i| - \max_i |z_i|]$

Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$. Define $$ F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big] $$ If $\|x\|_\infty \...
Drew Brady's user avatar
0 votes
0 answers
15 views

A question on Ibragimov's theorem on strong unimodality

I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
Ervand's user avatar
  • 51
1 vote
0 answers
35 views

Square-integral involving Brownian bridge

Let $B(t)$ be a standard Brownian bridge on $[0,1]$. Let $x>0$ be a (small) parameter. What is the distribution of $$ \int_0^{1-x} \left( B(t + x) - B(t) \right)^2 dt? $$ As noted I am interested ...
Kurisuto Asutora's user avatar
-1 votes
0 answers
25 views

Estimate the value of the PDF $P(f)$ at the minimal $f_0$ of the random-variable function $f(\mathbf{x})$

Let $f(\mathbf{x})=f(x_1,x_2,\dotsc,x_N)$ with $N>2$ be a real and continuous function and $f(\mathbf{x})\ge f_0$ for any $\mathbf{x}\in\mathbb{R}^N$. Now let $x_1,x_2,\dotsc,x_N$ be the i.i.d. ...
Guoqing's user avatar
  • 375
1 vote
0 answers
38 views

Can conditional distributions with respect to a sufficient sub-$\sigma$-algebra be represented by a single Markov kernel?

Let $(\Omega, \mathcal{F})$ be a measurable space, and let $\mathcal{P}$ be a collection of probability measures on this space. A sub-$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is said to be ...
MrTheOwl's user avatar
  • 111
3 votes
1 answer
139 views

Surjectivity of pushforward on image

Let $\mathcal X\subseteq\mathbb R^m$ be a Borel measurable set. $\Phi:\mathcal X\to\mathbb R^n$ be a continuous mapping and $\mathcal Y = \Phi(\mathcal X)\subseteq\mathbb R^n$ its image. Let $\mathcal ...
ECL's user avatar
  • 345
1 vote
0 answers
90 views
+100

Inequalities for norm of centered Gaussian and uncentered Gaussian

Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm. Let $x \in \mathbb{R}^n$ and define $$ F(x) = \mathbb{E}[\|x + g\| - \|g\|]. $$ I am wondering if it is possible to ...
Drew Brady's user avatar
1 vote
1 answer
54 views

Proving bound on expectation of likelihood ratio involving mixtures

Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
ILoveMath's user avatar
1 vote
0 answers
55 views

Quantitative multivariate CLT from quantitative CLT of linear combinations

Suppose $Z_1, \ldots, Z_k$ are random variables with mean $0$ and variance $1$ that are "approximately jointly Gaussian" in the sense that for any scalars $c_1, \ldots, c_k$, we have that $\...
Besfort's user avatar
  • 111
0 votes
0 answers
23 views

Characterisation of a family of continuous martingales

I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that $$X_0=0\quad \mbox{ and } \quad\...
Fawen90's user avatar
  • 1,389
0 votes
1 answer
82 views

Median of cardinality of set union

Let $U$ be an arbitrary finite universe (you can just think of it as $[N]=\{1,2,\ldots,N\}$), and $\mathbf{S} = (S_i)_{i \in [n]}$ ($S_i \subseteq U$) be the sets that we are drawing from. Define a ...
kingoyster's user avatar
3 votes
1 answer
84 views

What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?

Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having ...
evangecko's user avatar
0 votes
0 answers
37 views

Bounding the error of a truncated moment problem

Let $\{x_{i}\}_{i=1}^{\infty}$ be a non-increasing sequence of non-negative real numbers, and let $\{y_{j}\}_{j=1}^{B}$ be a non-increasing sequence of non-negative real numbers, where $B$ is a finite ...
CWC's user avatar
  • 433
-1 votes
0 answers
35 views

Different definition of Feller semi-group

(This is a crosspost of a question on MathStackExchange which did not receive any answer.) Let $E$ be a locally compact metric space, let $C_0(E)$ be the set of real-valued continuous functions of $E$ ...
Quiche_pro's user avatar
-1 votes
1 answer
93 views

Variance of bins for N balls into M bins [closed]

If I throw N balls independently into M bins with uniform probability, the expected mean of the M bins is N/M balls. What is the expected variance of the M bins? I was thinking of what bin size I ...
rationalfreak's user avatar
2 votes
1 answer
147 views

Lower bound in the singularity of random Bernoulli matrices

Let $A_n$ be a random $n \times n$ matrix with entries in $\{-1, +1\}$. As usual, "random" here means with respect to the uniform measure over such matrices. The strong version of the ...
Drew Brady's user avatar
3 votes
0 answers
80 views

Asymptotics of number of running maxima of iid random variables

Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities. Given a realisation $\omega$ of the random variables, we say that $X_i (\...
Nate River's user avatar
  • 6,155
5 votes
1 answer
375 views

Convergence of random functions

Suppose I have a sequence of random continuous functions, $f^{n} : [0, t] \to \mathbb{R}$. Suppose there also exists a random continuous function, $f: [0, t] \to \mathbb{R}$, defined on the same ...
Snidd's user avatar
  • 85
0 votes
0 answers
31 views

Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales

Does anybody know a reference for the following theorem? Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale. Then, for any constant $c > 0$, the event $(\exists > t)\, X_t \...
Neal Young's user avatar
1 vote
0 answers
79 views

Markov Chain that maximises the entropy creation rate

I am working on MERW (Maximal entropy random walk) for a project. I want to show that given a graph G, there is $\textbf{only one}$ aperiodic markov chain on G that maximises the entropy creation rate ...
ClaraS07's user avatar
4 votes
0 answers
116 views

Convergence in probability results with still open point-wise versions

In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
Matan Tal's user avatar
0 votes
0 answers
36 views

Contribution of Fisher information near jump points in convolved probability distributions

I am trying to compute the contribution to the Fisher information from jump points $b_i(\theta)$ in the convolved function $f(x; \theta)$ with respect to the parameter $\theta$. I am unsure whether it ...
Luna Belle's user avatar
9 votes
2 answers
429 views

Hermite–Fourier expansion for the median

Let $n$ be an odd positive integer. Let $M : \mathbb{R}^n \to \mathbb{R}$ be the median function: $M(x_1,\dots,x_n)$ is the median of $x_1,\dots,x_n$. What can be said about the Hermite–Fourier ...
Gil Kalai's user avatar
  • 24.7k
2 votes
1 answer
65 views

On the stationarity of Gaussian processes

I am trying to understand and prove the statement: The normal (or Gaussian) process is stationary in the wide sense if and only if it is strictly stationary. I know the following: A strictly ...
MathematicalMind1618's user avatar
3 votes
1 answer
219 views

Interpretation of an asymptotic result in probability

A result in asymptotic theory says the following: Let $Y$ be a real random variable with full support and $E(|Y|)<\infty$. Assume that: $$ (A)\quad \lim_{y\rightarrow \infty} \frac{\Pr(Y\geq y)}{\...
Star's user avatar
  • 108
3 votes
0 answers
81 views

Combinatorial/probabilistic interpretation of a quantity of union closed family

Let $\mathcal{F}\subseteq2^{[n]}$ be a union-closed family of sets. For a set $S\in[n]$ (not necessary belong to $\mathcal{F}$), define $w_{\mathcal{F}}(S)$ to be the number of subset of $S$ which ...
Veronica Phan's user avatar
0 votes
1 answer
66 views

Does convergence in probability of iid samples imply convergence in measure of the sampled functions?

Let $g_i: [0, 1] \to \mathbb R$ be $L^1$ functions, equibounded in $L^1$ norm. Let $X_i$ a sequence of iid uniform random variables on $[0, 1]$. Suppose that $$\frac{1}{n} \sum_{i = 1}^n g_i (X_i) \to ...
Nate River's user avatar
  • 6,155
3 votes
1 answer
405 views

Moments of a random variable related to uniform distribution on sphere

Let $u$ be taken uniformly from the unit sphere $\mathbb S^{n-1}$ and $D$ be a diagonal matrix. I'd like to find a general formula for $$ \mathbb E[(u^\top D u)^m] $$ for $m=1,2,3, \dots$, in terms of ...
Pluviophile's user avatar
  • 1,608
-3 votes
0 answers
135 views

Approximation on Dirichlet's arithmetic progression by means of central limit theorem

In this video lecture on Number theory over function fields taught by Will Sawin is presented a 'conceptional' reason for error estimation $\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \} =\frac{1}...
JackYo's user avatar
  • 619
0 votes
2 answers
126 views

Unique coupling

Let $X$ be a Polish metric space, and let $\mu,\nu$ be two Borel probability measures on $X$, when is the product measure the only coupling of $\mu$ and $\nu$. More formally, let $$\Gamma(\mu,\nu):=\{\...
Andrea Aveni's user avatar
1 vote
0 answers
91 views

How to optimize parametric information-theoretic bounds?

I am faced with an information-theoretic upper bound, such as \begin{align} \sqrt{\alpha'}2^{I_\alpha(X;Y)}, \end{align} where $I_\alpha(X;Y)$ is the Rényi mutual information with parameter $\alpha>...
Math_Y's user avatar
  • 287
-2 votes
0 answers
52 views

Density of squared bessel process

I was trying to find a transition density function for a squared Bessel process. In the book "Continuous martingale and Brownian motion" by Revuz and Yor, I find a Corollary on page 441 that ...
LOREY CHU's user avatar
-1 votes
0 answers
27 views

Number variance of random points (and deviations for empirical processes)

Let $X_1, X_2, \dots$ be i.i.d. random variables having uniform distribution on $[0,1]$. Write $I_{t,x}$ for the indicator function of an interval of length $x$ with center $t$. Consider $$ V(N,x) = \...
Kurisuto Asutora's user avatar
0 votes
1 answer
72 views

Lower Bound on the Probability for the Sum of IID Random Variables

Let $X_1,\ldots,X_n$ be $n$ iid normalized random variables (with finite variance, possibly sub-Gaussian). Suppose further that $\mathbb{P}(X_1 > 0 ) > 1/2$, implying a positive skew in the ...
xabialgebra's user avatar
8 votes
1 answer
534 views

The cars problem, again

Consider the following simple problem: We are given $2n$ parking spots, labelled from 0 to $2n-1$. There are $n$ cars on the first $n$ spots, and the remaining $n$ spots are free. At every step, every ...
AccidentalFourierTransform's user avatar
1 vote
0 answers
58 views

Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)

Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation: $$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
user1172131's user avatar
0 votes
1 answer
95 views

On the behaviour of individual random walks of a Markov Chain

My current research (on Probabilistic Automaton) brought me to the following question regarding Markov Chains. I state the definitions for the sake of clarity. Let $M$ be a discrete-time finite Markov ...
santi cifu's user avatar
3 votes
0 answers
90 views

Tighter Freedman's inequality for a special martingale difference sequence

Let $X_{1}, \ldots, X_{T} \in \{0, 1\}$ be a sequence of Boolean random variables with $$ \mathbb{E}[X_{t} | X_{1}, \dots, X_{t - 1}] = p_{t}. $$ Consider the sequence $Y_{t} := X_{t} - p_{t}$ (which ...
Fellow4's user avatar
  • 41
0 votes
1 answer
51 views

Reconstruction of law of diffusion process from call option values

Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the $$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$ Then, ...
ABIM's user avatar
  • 5,405