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.
8,631
questions
2
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Is independence preserved if a random entry in an independent sequence is replaced by a constant?
Let $\xi=(\xi_1,\ldots,\xi_n)$ be a sequence of independent random variables.
Let us pick an index $\nu\in \{1,\ldots,n\}$, and replace the entry $\xi_\nu$ by a constant $c$. The rest of the $\xi_i$ ...
2
votes
1
answer
396
views
If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set
If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
2
votes
0
answers
56
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Given a set of marginals, what is the largest support of a distribution satisfying these?
Given a random variable $X$ with support over $\{0,1\}^I$, we can define the marginal distribution on the bits indexed by $A \subseteq I$ by $Pr(X_A = x_A) = \sum_{x \in \{0,1\}^{I - A}} Pr(X = x \cup ...
1
vote
1
answer
3k
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Correlation between square of normal random variables
Suppose I have $X,Y$ bivariate normal with correlation coefficient $\rho \in (0,1)$ . Then , what is the correlation between $X^2 $ and $Y^2$ ?
I am aware of the fact that the square of the normal ...
3
votes
2
answers
762
views
Weak convergence of conditional probabilities
Suppose $\mu_n\implies\mu$, i.e. $\mu_n$ converges weakly to $\mu$ where $\mu_n$, $\mu$ are probability measures on some metric space $(X,d)$. Given a Borel set $B$, define $\mu^B$ to be the ...
2
votes
1
answer
838
views
Comparing mixing time of lazy and non-lazy Markov chains
Suppose we have a probability distribution $\pi : X \rightarrow [0,1]$ where $X$ is finite and let $Q : X \times X \rightarrow [0,1]$ be a Markov kernel that is reversible with respect to $\pi$. That ...
4
votes
2
answers
2k
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Bounds on the mills ratio
How do I show the following bounds on the mills ratio :
$\frac{1}{x}- \frac{1}{x^3} < \frac{1-\Phi(x)}{\phi(x)} < \frac{1}{x}- \frac{1}{x^3} +\frac{3}{x^5} \ \ \ \ \ \ \ $ for $ \ \ \ x>0$ ...
5
votes
1
answer
257
views
Find a function $F$ on $[0,1]$ with moments decaying as $(\ln n)^{-n}$
Let $F:[0,1]\to\mathbb{R}$ be a measurable function such that
$$
\mu_n(F)=\int_0^1F(t)t^ndt\sim\frac1{(\ln n)^n}\quad\mbox{as}\quad n\to\infty.
$$
More precisely,
$$
0<c<|\mu_n(F)|(\ln n)^n<...
8
votes
0
answers
151
views
Pursuit-evasion with many slow pursuers
Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...
3
votes
2
answers
301
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Lower bound Renyi divergence between two discrete probability distributions
I am trying to understand the proof of Lemma 1 in this paper (Section 9.2).
The proof shows that given a discrete probability distribution $P=(p_1,p_2,...,p_k)$ where $p_1 \geq p_2 \geq ... \geq p_k$,...
4
votes
1
answer
120
views
Different type of measurability of transition kernel
Let $(E,d)$ be a Polish space equipped with the Borel $\sigma$-algebra $\mathcal{E}$. Let $\mathcal{P}(E)$ be the space of all probability measures on $(E,\mathcal{E})$. We eqiup this space with the ...
2
votes
0
answers
56
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Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$
Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try...
So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...
1
vote
1
answer
41
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Probability that maximal elements has the same position in samples from correlated random variables
Let $x$ and $y$ be two correlated random variable (say, standard normal) with correlation coefficient $\rho>0$. Let $X= \{x_1, x_2, ..., x_L\}$ and $Y= \{y_1, y_2, .. y_L\}$ be samples of size $L$ ...
2
votes
1
answer
226
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Kolmogoroff condition for truncated random variables
Question summary. Does the Kolmogoroff condition $\sum_{n=1}^\infty\frac{\mathbb V Y_n}{n^2} < \infty$ hold for truncated random variables $Y_n := X_n \cdot 1_{\{X_n \le n\}}$ (see below for a more ...
3
votes
1
answer
194
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Quantitative CLT bound
Consider an independent collection of random variables $W_i, i=1,\dots,n.$ and let $Z \sim N(0,1)$. Roughly speaking, we know that $W_i$ are close in distribution to $Z$, say each is itself a sum of $...
8
votes
2
answers
618
views
Does a random sequence of vectors span a Hilbert space?
Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...
3
votes
0
answers
173
views
Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion
I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...
5
votes
1
answer
165
views
Comparison of Rademacher processes
Suppose that $T$ is a bounded set in $\mathbb{R}^n$ and $f,g$ are two nonnegative functions such that $0\leq f(x)\leq g(x)$ for all $x\geq 0$.
Let $\epsilon_1,\epsilon_2,\dots,$ be a Rademacher ...
1
vote
0
answers
113
views
Sanov-type finite-sample bound on $KL(P\|\hat{P}_n)$
Let $P$ be a distribution on an alphabet of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ via $n$ i.i.d samples $a_1,\ldots,a_n \sim P$, i.e $\hat{P}_n := (1/n)\sum_{i=1}^n\delta_{a_i}$.
...
0
votes
1
answer
91
views
If a strong Markov process reaches a Borel set a.s., can it be restarted from that set?
Let $X$ be a strong Markov process on $E$, and $B\in \mathcal B(E)$. Suppose that, for some $x\in E$,
$$
P_x(\exists t\ge0 \text{ such that } X_t\in B)=1.
$$
My question: Does there exist a stopping ...
1
vote
1
answer
72
views
Independent identical distribution sequence
given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $.
I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \...
1
vote
1
answer
289
views
Bounds on difference between "logsumexp" and variance?
Let $Z$ be a random variable with finite moment-generating function $M_Z(\theta):=E[e^{\frac{1}{\theta}Z}]<\infty$ for all $\theta > 0$, and for $\delta \in (0,1]$, define
$C_Z^\delta := \inf_{\...
0
votes
2
answers
5k
views
Quadratic covariation of two not independent Brownian motions
Given two not independent Brownian motions, $X$ and $Y$. I was wondering if we can say anything about the quadratic covariation of $X$ and $Y$, $\langle X,Y \rangle_t$. I know that for two independent ...
1
vote
0
answers
150
views
Does the law of a Feller process depend continuously on the initial condition?
Let $E$ be a locally compact and separable metric space, and suppose $X$ is a Feller process with transition function $P_t$. To be precise, let $C_0$ denote the space of continuous functions vanishing ...
1
vote
0
answers
172
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Two mixing rates of random dynamical system
Given random dynamical system $(X, \mathcal{B}, (T_{\omega})_{\omega\in \Omega}, \mu)$ where $(\Omega, \mathbb{P})$ is probability space with ergodic transformation $\sigma: \Omega \to \Omega$. Define ...
2
votes
2
answers
235
views
Substitute Concrete Value in Conditional Expectation
Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space.
Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables.
Furthermore, let
$$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$
be a $...
2
votes
0
answers
95
views
Smallest singular value distribution
Let $G_\mathbb{R}\in\mathbb{R}^{n\times n}$ and $G_\mathbb{C}\in\mathbb{C}^{n\times n}$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian ...
5
votes
3
answers
464
views
Optimisation under constraint of Wasserstein distance
Let $\mathcal P_n = \{P \in \mathbb R^n_{\geq 0}: P^T \mathbb I = 1 \}$, where $\mathbb I = (1,...,1)^T \in \mathbb R^n$ and $f: \mathcal P_n \to \mathbb R$ a convex and differentiable function (or ...
1
vote
1
answer
89
views
The weak version of the memoriless property
In our group we are working with a probability distribution $X$ defined on a non-negative domain, satisfying the following property
$$
P\left[X>a\right]\ge P\left[X>a+t \mid X>t\right],
$$
...
8
votes
1
answer
311
views
Lower Bound of KL-Divergence Between Two Gibbs Measures
Suppose we have two Gibbs measures with densities
$$
p_f(x) \propto \exp(f(x)),\quad q_g(x)\propto \exp(g(x)).
$$
Consider the KL-divergence between $p_f$ and $q_g$, as a functional of $f$ and $g$, ...
1
vote
0
answers
58
views
Regularity of the pdf of partial Birkhoff sums
Suppose that $T: X \to X$ is some measurable map on a Riemannian manifold $X$ (possibly with boundary). Let $\mu$ denote the Riemannian measure on $X$. For measurable, real-valued $g$ we may consider ...
2
votes
0
answers
69
views
If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\...
1
vote
1
answer
711
views
On Riemann integration of stochastic processes of order $p$
Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
5
votes
2
answers
176
views
Density near at $0$ for the integral of the positive part of the Brownian motion
This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...
2
votes
1
answer
145
views
Difference of two probability measures modulo a third
Given three probability measures on $N$ elements (so $\mu_0, \mu_1,\mu_2 \in \ell^1_N$), I need to define the difference of $\mu_1$ and $\mu_2$ "modulo" $\mu_0$ as
$$
\sup \bigg\{ \int f \,\mathrm{d}(...
1
vote
1
answer
158
views
Strong convexity of internal energy with respect to Wasserstein metric
It is well known that the internal energy (see, e.g., Definition 3.32 in and Proposition 3.33 in 1) is geodesically convex with the 2-Wasserstein distance. I was wondering under what condition, the ...
1
vote
0
answers
101
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A question about pdfs with likelihood ratio order
Suppose $f_1,f_2,\dots$ are pdfs of absolutely continuous random variables with the same support (say an interval). Assume that $\{f_i\}$ are strictly positive in their support. Furthermore, $\frac{...
3
votes
1
answer
3k
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Is there a tight lower bound for the expectation of the product of two positive valued random variables?
Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.
I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.
...
2
votes
1
answer
209
views
Smoothness of expectation
Suppose that $X_t$ is a strong solution to the SDE,
$$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...
1
vote
1
answer
92
views
boundig variation from median [closed]
Given a scalar random variable $X$, suppose that there are positive constants $c_{1}$ and $c_{2}$ such that
$$\forall t\geq 0 : \,\,\,\,\,\,\ \mathbb P\{|X-\mathbb EX|\geq t\}\leq c_{1}e^{-c_{2}t^{2}}...
1
vote
1
answer
154
views
Find probability of non-stationary inputs into Turing machine?
Consider some finite string $x=(x_1,x_2,...,x_{n-1},x_n)$ that is drawn from a non-stationary process. Would it be possible to use the algorithmic probability formula, defined by Solomonoff as,
$$
P_M(...
1
vote
2
answers
218
views
L1 distance after Convolution
Given two discrete distributions $P$ and $Q$ with the same support $x_1,\cdots,x_n$. Assume $K \in L^1(\mathbb{R})$ is a nonnegative function with $\int_\mathbb{R} K(x)dx = 1$, and let $K_h(x) = \frac{...
4
votes
1
answer
146
views
Large Deviations for Self-Normalized Sums
I am trying to understand the main result (Theorem 1.1) in this paper by Shao, which gives a large deviation bound for the self-normalized sum of iid variables
$$
\frac{\sum X_i}{\sqrt{n}\sqrt{\sum ...
1
vote
0
answers
78
views
What is the Bruss-Yor concept of no information?
A few years ago, a question related to a paper of Thomas Bruss and Marc Yor on the so-called last arrival problem received some attention on this forum.
What I'd like to know now is:
What are the ...
7
votes
1
answer
434
views
What numbers can simulate 1/2?
Given two numbers $p,q\in(0,1)$, we say that $p$ can simulate $q$ if, given a biased coin with probability $p$, we can toss it a bounded number of times and use the results to simuate a biased coin ...
4
votes
0
answers
141
views
Local behaviour of fractions with bounded denominator / Was it already studied?
My question is about a point process that I feel it would be natural to study, but that I have never heard of… This point process would represent, morally, the local behaviour of the set of fractions ...
0
votes
0
answers
128
views
Knights on an n x n chessboard
In a n x n chessboard a white knight sits on the top left corner, and a black knight on the bottom right corner. Starting with white, the two knights take turns to move at random, and with equal ...
9
votes
1
answer
460
views
What does convergence in distribution "in the Gromov–Hausdorff" sense mean?
I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m_n,d_n) \to (m_{\infty}, d_{\infty})$$
"...
1
vote
1
answer
104
views
What is the distribution of a Cartesian power of a collection of iid uniform points? (renewed)
The following question was asked recently at https://mathoverflow.net/questions/326631/what-is-the-distribution-of-a-cartesian-power-of-a-collection-of-iid-uniform-poi :
Take a rectangle with ...
63
votes
2
answers
3k
views
Guessing each other's coins
I recently thought about the following game (has it been considered before?).
Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an ...