All Questions
Tagged with pr.probability inequalities
38 questions
20
votes
2
answers
6k
views
Constants in the Rosenthal inequality
Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $X = \sum_{i=1}^n X_i$. Then we have the family of "Rosenthal-type ...
- 201
1
vote
1
answer
143
views
Comparison of hitting probability of two Markov chains both with only one absorbing state version 2 under stronger condition
Let $N_n:=\{1,2,\cdots,n\}$. Given two finite states Markov chains $\big(X^{(j)}_i\in N_n\}\big)_{i=0}^\infty$ for $j\in\{1,2\}$, both of which have two absorbing states at $1$ and $n$.
$\text{Pr}\...
- 2,239
37
votes
3
answers
3k
views
An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
- 11.4k
36
votes
3
answers
4k
views
the following inequality is true，but I can't prove it
The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...
- 363
26
votes
3
answers
3k
views
An $L^0$ Khintchine inequality
Suppose that $\epsilon_1,\epsilon_2,\ldots$ are IID random variables with the Bernoulli distribution $\mathbb{P}(\epsilon_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ is a real sequence with $\sum_na_n^2=1$. ...
- 17.1k
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
18
votes
3
answers
3k
views
Entropy and total variation distance
Let $X$, $Y$ be discrete random variables taking values within the same set of $N$ elements. Let the total variation distance $|P-Q|$ (which is half the $L_1$ distance between the distributions of $P$ ...
- 20.2k
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 178
11
votes
1
answer
676
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 196
10
votes
3
answers
4k
views
Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability)
Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is
$$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$
...
- 223
7
votes
1
answer
499
views
Moments of a random variable and of its conditional expectation
Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested ...
- 1,269
6
votes
1
answer
375
views
Deviation bound for the maximum of the norm of Wiener process
Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof:
$$
{...
- 61
5
votes
4
answers
4k
views
Is there an inequality relation between KL-divergence and $L_2$ norm?
According to the Pinsker inequality, we have the following inequality:
\begin{equation}
\delta_{TV} (p, q)^2 \leq \frac{1}{2} D_{KL}(p,q),
\end{equation}
where $\delta_{TV} (\cdot, \cdot)$ and $D_{KL}...
- 175
5
votes
0
answers
295
views
inequality in a shape of inclusion exclusion formula
I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality:
consider 9 numbers $a_1,a_2,...
- 341
5
votes
2
answers
202
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
- 128k
5
votes
1
answer
512
views
Concentration inequality for Hilbert space valued random variables
I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
- 115
4
votes
3
answers
161
views
Find distribution that minimises a function of its moments
Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive).
How can I find $f(x)$ that ...
- 223
4
votes
1
answer
168
views
Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound
Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound.
Is there a ...
- 143
4
votes
1
answer
285
views
Probability of existence of $\lambda$ such that $\lambda a_i \geq b_i$, for i.i.d random variables $a_i$'s and $b_i$'s
Suppose $a_i$'s and $b_i$'s ($1\leq i\leq n$) are i.i.d Gaussian random variables. What's the probability that a $\lambda$ exists such that $\lambda a_i \geq b_i, ~\forall i$?
Actually, an upper bound ...
- 1,216
4
votes
1
answer
863
views
Hoeffding's inequality for Hilbert space valued random elements
Suppose that $\mathbb H$ is a separable Hilbert space and $X_1,\ldots,X_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|...
- 423
4
votes
2
answers
2k
views
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$ ...
- 183
4
votes
1
answer
474
views
Concentration inequalities in $\ell_{\infty}$ for sums of iid random ("nice") functions?
I'm looking for "tail-bound-like" inequalities that look like this (I state a specific setting but more general settings are interesting):
Let $D$ be a distribution on a set of "nice" functions $g$:...
- 4,529
4
votes
1
answer
1k
views
Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences
A classic formulation of the Bernstein inequality (from Wikipedia) is as follow:
Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$...
- 53
3
votes
1
answer
142
views
How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?
Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$,
$$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
user478492
3
votes
1
answer
197
views
possibility of bounding one functional by another functional
This is a natural follow-up question related to one of my previous questions at here. Assume that $\rho$ is a log-concave probability density function with support $[0,\infty)$ and fixed mean $\mu >...
- 730
3
votes
1
answer
3k
views
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.
...
3
votes
0
answers
494
views
Maximization of a total variation distance subject to another total variation distance in Markov chain
Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
- 1,109
3
votes
1
answer
229
views
Inequality for difference of consecutive atom probabilities for binomial distribution
Edit: This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its ...
- 2,720
2
votes
1
answer
136
views
Does higher volatility of SDE imply lower probability of staying positive?
Given two SDEs $X^1$, $X^2$ :
$$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$
where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$....
- 1,334
2
votes
1
answer
796
views
Can we get that $ P(N^{2/3}(\lambda_N-\lambda_{N-1})\le c)\ge 1-\epsilon$?
Following this question: Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law?.
We know that
$$
\lim_{N\to\infty}P(N^{2/3}(\lambda_N-2)\le s_1,\dotsc,...
- 288
2
votes
3
answers
222
views
Can we find such $k$ so that the following inequality holds?
I found this question: Chernoff style concentration bound for ratio of variables.
I want to ask if we get similar thing for the ratio of the sum and the one Gaussian variable.
Given i.i.d. Gaussian ...
- 288
2
votes
0
answers
124
views
Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?
Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
- 5,470
2
votes
1
answer
161
views
Trying to bound one functional by another functional
In my little research project, I faced the following problem: Assume that $\rho$ is a probability density function with support $[0,\infty)$ and mean $\mu >0$. Let $$H[\rho] = \iiint_{y,v,w\geq 0} \...
- 730
1
vote
1
answer
97
views
A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$?
To get the non-strict version of ...
- 128k
1
vote
1
answer
227
views
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it ...
- 3,477
1
vote
0
answers
133
views
Does the Gaussian Poincare inequality hold for infinite dimensional measure metric spaces?
This is a question subsequent to the one:
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
There, I received a very helpful answer that the Gaussian poincare inequality for any ...
- 3,477
1
vote
1
answer
362
views
An inequality involving the Wasserstein distance and chi-squared distance
$\newcommand{\N}{\mathbb N}$Let $P$ be the set of all probability mass functions on $\N_0:=\{0\}\cup\N$, where $\N:=\{1,2,\dots\}$. Let $P_{>0}$ denote the set of all $q=(q_0,q_1,\dots)\in P$ such ...
- 128k
0
votes
1
answer
503
views
Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector
Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral
$I(...
- 968