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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.

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Is the probability of the Euler-Mascheroni constant being rational large? [closed]

Let $\gamma$ denote the Euler Mascheroni constant. Show there exist infinitly many polynomials $p\in \mathbb{Z}[x]$ such that $p(x) = \gamma$. Further, show one can calculate a probability that some ...
John Wayne's user avatar
6 votes
0 answers
100 views

Size doubling amoeba

Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$. A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p &...
Nate River's user avatar
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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
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1 vote
0 answers
79 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
3 votes
0 answers
70 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
18 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
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1 vote
0 answers
45 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
26 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
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0 answers
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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
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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
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2 votes
0 answers
97 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
56 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
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0 answers
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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
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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
85 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
38 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
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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
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5 votes
1 answer
376 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
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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
81 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
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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
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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
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-3 votes
0 answers
136 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
92 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
1 vote
0 answers
42 views

Sub-Gaussian analysis via bounded decomposition?

Let $\psi_\alpha(x) := \exp(x^\alpha)-1$. The Sub-Gaussian Norm $\lVert X \rVert_{\psi_2}$ of a random variable $X$ is defined as $$ \lVert X\rVert_{\psi_2} = \inf\{c>0\mid \mathbb{E}[\varphi_2(|X|/...
Mark Schultz-Wu's user avatar
0 votes
2 answers
116 views

Upper bounds on quotients of binomial coefficients

Let $\gamma>1$ be a real number and let $n\in \mathbb{N}$. Define $f\colon\mathbb{N}\to[0,1]$ $$ f(n_0) = \frac{\binom{n-n_0}{m}}{\binom{n}{m}}, $$ where $$ m = \Big\lfloor{\frac{n}{\lceil\gamma ...
xabialgebra's user avatar
1 vote
1 answer
51 views

How do the total variation distances of the marginals relate to the total variation distance of the joint under independence?

Suppose there are two sets of random variables $X_1,...,X_n$ and $Y_1,...,Y_n$ with all the variables being defined over the same sample space, but not necessarily being identically distributed. Is ...
David Pascall's user avatar
3 votes
1 answer
158 views

Sub-Gaussian concentration without the sub-Gaussian norm

A random variable $X$ is said to have sub-Gaussian tails with parameter $\sigma>0$ if $$\Pr[|X|\geq t] \leq 2\exp(-t^2/(2\sigma^2))$$ I am interested if $X_0, X_1$ are independent, and have sub-...
Mark Schultz-Wu's user avatar
4 votes
1 answer
227 views

Problem in Probability Theory and Functional Analysis

Let's consider the vector space V of bounded scalar functions, which includes the constant function 1. We assume that any uniform limit of a bounded monotonic sequence of functions from V also ...
Nasim Mamatkylov's user avatar
1 vote
1 answer
150 views

Resource request (probability theory, computability theory, algebra)

I'm a first year graduate student trying to explore specific topics I might be interested in researching. Currently, I enjoy algebra, probability theory, and the computability theory side of logic, ...
modz's user avatar
  • 121
3 votes
2 answers
282 views

Nash equilibria of a "minority game"

An odd number $N \geq 3$ of players are playing a game - they bet on the outcome of a biased coin that comes up heads $p > \frac{1}{2}$ of the time, where $p$ is known to all of the players in ...
Nate River's user avatar
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