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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
3 answers
160 views

Upper Bound for $\mathbb{E}\left[\max_{j \in \mathcal{N}} h_{j}\right]$

Assume $\{h_j\}_{j\in \mathcal{N}}$ are independent Gamma random variables, each with potentially different distributions and parameters. I am looking for an upper bound for $\mathbb{E}\left[\max_{j \...
Lee White's user avatar
7 votes
0 answers
263 views

$\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]$ where $W$ and $V$ are independent

Let $V$ and $W$ be independent random variables. Assume that $V$ is standard normal. We are interested in the following limit \begin{align} \lim_{n \to \infty} E[V| W +\frac{1}{n}V ] \end{align} ...
Boby's user avatar
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2 votes
1 answer
154 views

strict inequality for Fatou's lemma

It is not the well-known form of Fatou's lemma. It is shown as below: let $g\ge 0$ be continuous. If $X_n$ weakly converge to $X$ then $$\lim\inf_{n\rightarrow \infty} Eg(X_n)\ge Eg(X)$$ I'd like to ...
Zhuozhen Jiang's user avatar
16 votes
3 answers
782 views

Show there is no positive r.v. $U$ such that $\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge (k+1)/2 }]}{\mathbb{E}[U^k]}, \, \forall k \in \mathbb{N}_0$

Let $U$ be a non-negative random variable such that for all $k \in \mathbb{N}_0$ \begin{align} \frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge \frac{k+1}{2} }]}{\mathbb{E}[U^k]}. \end{align} In ...
Boby's user avatar
  • 671
-1 votes
1 answer
80 views

Seating assignment inspired question

Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course ...
Dominic van der Zypen's user avatar
2 votes
0 answers
71 views

Assumptions Wald's second equation?

Let $(X_n)_{n\in \mathbb{N}}$ be an i.i.d. sequence of random variables and $N$ an $\mathbb{N}_0$ valued random variable. Let $X_1 \in \mathcal{L}^2$ and $N \in \mathcal{L}^1$. Let $S_n := \sum_{i=1}^...
psl2Z's user avatar
  • 261
16 votes
0 answers
309 views

Randomized Pascal's triangle: What is the average of all the numbers?

This question was posted on MSE. It received some interesting responses, but no definite answer. Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for ...
Dan's user avatar
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1 vote
0 answers
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Equality regarding score function

$\newcommand{\Var}{\operatorname{Var}}$I'm reading Bogetoft and Otto (2011) page 249 and this equality regarding the score function stumped me: $$\Var\left[ -\Big(\frac{\partial^2 l}{\partial \beta^2}\...
Shay's user avatar
  • 11
0 votes
0 answers
133 views

Asymptotics of a ratio on the unit sphere

Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$. Consider the ratio (for $k \geq n$) $$ R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-...
Drew Brady's user avatar
6 votes
0 answers
164 views

Stopping time on the sum of a moving window

(Cross post from MSE) Let $\xi_i$ be iid random variables, and define: $$S_{(k)} = \sum_{i=1}^N \xi_{i+k}$$ Now, define: $$\tau = \min \left\{ k : S_{(k)} \notin (a,b) \right\}$$ How can I find $\...
user3141592's user avatar
0 votes
1 answer
45 views

Expectation of spectral norm of a diagonal stochastic matrix

There is a diagonal matrix $G(t)\in R^{M\times M}$ where the diagonal elements are independent Bernoulli stochastic variables, satisfying $\mathbb{E}(g_i(t) = 1) = b$, and $ \mathbb{E}(g_i(t) = 0) = 1 ...
Keven Sun's user avatar
2 votes
1 answer
73 views

Connection between Wassertein-2 metric and difference in variance

Given two probability densities $\mu\in\mathcal P(\mathbb R^d)$ and $\nu\in\mathcal P(\mathbb R^d)$, we define their Wasserstein-$p$ metric as $$ W_p^p(\mu, \nu)=\inf_{\gamma\in \Gamma(\mu, \nu)}\int_{...
Daniel Cortild's user avatar
0 votes
0 answers
37 views

Expected (log)volume of projection of a cube onto a random subspace

Suppose $A$ is a full-rank $d\times d$ dimensional matrix. Let $U \in R^{d\times k}$ be a projection to projection matrix onto a uniformly chosen sub-space of dimension $k$ (for example, they can be ...
kvphxga's user avatar
  • 187
1 vote
0 answers
36 views

Uniform distribution as argument for copula likelihood

I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...
Grigori's user avatar
  • 33
1 vote
1 answer
111 views

Expected number of solutions of a random quadratic polynomial system over a finite field

Let $\mathbb{F}_q$ be a field of $q$ elements. Let $a_{i,j,k}$, $b_{i,j}$, $c_i$ ($1 \leq i \leq m$, $1 \leq j \leq k \leq n$) be independent uniformly distributed random variables in $\mathbb{F}_q$, ...
en-drix's user avatar
  • 157
3 votes
1 answer
271 views

Expectation on a Polish space

I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral $\int_X x dp$ ...
J.R.'s user avatar
  • 291
-1 votes
1 answer
503 views

Minimum of exponential distribution

Consider $n$ independent random variables $𝑋_𝑖\sim\exp(𝜆_𝑖)$ for $I=1\ldots,n$. Let $\lambda = \sum_{i=1}^n\lambda_i$. Of course, the minimum of these exponential distributions has distribution: $...
Jim Chen's user avatar
1 vote
0 answers
107 views

Expected value of maximal cycle length in fixed-point free bijections

$\newcommand{\n}{\{1,\ldots,n\}}$ $\newcommand{\FF}{\text{FF}}$ $\newcommand{\lc}{\text{lc}}$ Motivation. A group of my son's peers decided to have a few days of Secret Santa before last year's ...
Dominic van der Zypen's user avatar
3 votes
1 answer
211 views

Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process

Consider the modified Ornstein–Uhlenbeck process $$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$ for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
Jean Daviau's user avatar
9 votes
1 answer
788 views

The expected value of product of random variables which have the same distribution but are not independent

Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\prod_{i=1}^k X_i\right)\geq c(k)$ for any $k$-random variables $X_1,X_2,\ldots,X_k$ which all have the ...
fengzju's user avatar
  • 99
14 votes
1 answer
1k views

A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?

A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points. For example, here are $20$ random ...
Dan's user avatar
  • 3,507
-1 votes
1 answer
148 views

Weighted sum of zero-mean random variables

Let us say we have two independent random variables $X$ and $Y$, with both $E[X] = E[Y] = 0$. Is it true that for any random weight variable $0 \le W \le 1$ (e.g., $W$ dependent on $X$ and $Y$) we ...
Yauhen Yakimenka's user avatar
8 votes
3 answers
595 views

Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$

Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution. What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
TheSimpliFire's user avatar
0 votes
2 answers
239 views

Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices

I am working with two random matrices, $Z$ and $H$: $Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$. $H$ is a $K \times K$ ...
Dalek's user avatar
  • 37
0 votes
0 answers
46 views

Prove lower collinearity on the tails of Gaussian blob

Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
user1172131's user avatar
2 votes
0 answers
74 views

Expected number of steps until a queue of $n$ people has passed all $n$ ordered tests consecutively

We are given a queue of $n$ people $\{p_1, \ldots, p_n\}$. They each have to pass $n$ exams $\{t_1, \ldots, t_n\}$. For simplicity we can "draw" the setting in the following way: $$[t_n,t_{n-...
Lucas's user avatar
  • 21
5 votes
1 answer
466 views

Bounding expected maximum via adjacent differences

For $X_i$, $i\in[n]$ be a sequence of integrable random variables. Is there a universal constant $c>0$ such that $$\mathbb{E}\max_{i\in[n]}X_i \le c\left( \max_{i\in[n]}\mathbb{E}|X_i| + \mathbb{E}\...
Aryeh Kontorovich's user avatar
5 votes
2 answers
165 views

Expectation of a function of two entries of an isotropic unit vector $\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\![{w_{1}}^{\!4}\,{w_{2}}^{\!4}]$

Problem: Given a random isotropic unit vector in $\mathbb{R}^p$ for $p\ge2$, we are trying to compute (preferably exactly, otherwise to upper bound): $$\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\!\...
Itay's user avatar
  • 673
0 votes
1 answer
113 views

Inequality on conditional variance of a vector

I have a random vector $X$ and an event $\mathcal{E}$ such that $\mathbb{P}(\mathcal{E}) = p$. I am trying to show the following inequality : \begin{equation} p\mathbb{E}[\|X - \mathbb E [X \vert \...
karel's user avatar
  • 11
0 votes
0 answers
84 views

Expectation of the trace of random matrix with an inverse insided

Consider a $N$-dimensional random complex vector $\mathbf{x} \in \mathbb C^{N \times 1}$ following the complex Gaussian distribution, i.e., $\mathbf{x} \sim {CN}(0,\sigma^2 \mathbf{I})$, where $\...
Prokins Wang's user avatar
0 votes
1 answer
324 views

Finding examples of functions which are infinite or undefined with current extensions of the expected value?

Preliminaries Consider the expectations desribed in this paper, which is an extension of the Lebesgue density theorem; this paper which is an extension of the Hausdorff measure, using Hyperbolic ...
Arbuja's user avatar
  • 63
2 votes
0 answers
122 views

Are the extensions of the expected value, below, finite for all functions in only a shy subset of all measurable functions?

This is a follow up to this post, where I wish to verify whether one of the statements (in the post) is true but first let's recap the definitions: Let $(X,d)$ be a metric space. If set $A\subseteq X$,...
Arbuja's user avatar
  • 63
6 votes
2 answers
291 views

Expectation of the inner product of a subset of two random orthonormal vectors

Setting: Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, ...
Itay's user avatar
  • 673
1 vote
0 answers
739 views

Finding a unique and finite expected value for almost all measurable functions?

Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...
Arbuja's user avatar
  • 63
13 votes
2 answers
398 views

Expected sorting time of random permutation using random comparators

In sorting networks, a comparator of positions $i < j$ is an operator which takes a permutation, checks if $p_i > p_j$, and if it is the case, swaps $p_i$ and $p_j$. Using this, we can define ...
Daniel Weber's user avatar
  • 3,319
0 votes
1 answer
94 views

Convergence in expectation of a discontinuous function

Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...
dhp's user avatar
  • 11
2 votes
1 answer
194 views

Bound the probability that a point belongs to a set

Let $(a_k)_{k \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $$ (1) \quad \Pr\Big (\lim_{n\rightarrow +\infty}d(\frac{1}{n}\sum_{k=1}^n 1_{[a_k = b]}, [v_\ell(b,\...
Star's user avatar
  • 108
0 votes
1 answer
196 views

Bound the expectation of an average

Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...
Star's user avatar
  • 108
2 votes
1 answer
196 views

Tossing a coin around $\mathbb{Z}/n\mathbb{Z}$ [closed]

Motivation. With my younger son I played the following game on a big (dysfunctional) clock which can be modelled as $\mathbb{Z}/12\mathbb{Z}$ : Put the clock hands at number $12 ( = 0)$. Toss a coin, ...
Dominic van der Zypen's user avatar
1 vote
1 answer
204 views

Expected (maximum minus minimum) of Laplacian random variables

Suppose there are $n$ IID random variables denoted as $X=(X_1,\dots, X_n)$, they follow Laplace distribution with parameter $\lambda$, denoted as $Lap(\lambda)$. That is, $$f(x)=\frac{1}{2\lambda}\exp ...
white's user avatar
  • 23
2 votes
1 answer
324 views

On the mean value taken by Bernoulli random variables with joint distribution constraints

We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X_1, X_2, \ldots X_n$, such that the probability $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...
Penelope Benenati's user avatar
0 votes
1 answer
115 views

Approximation for an expectation expression

Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
A. R.'s user avatar
  • 25
2 votes
0 answers
42 views

can we get a family of classifiers $\left\{f_n\right\}_{n \in N}$such that $\lim_{n->∞} (E_{(X_1, Y_1), ...,(X_n, Y_n) \sim \rho}[R(f_n)]-R(f_B))=0 $

For a given classifier $f: \mathbb{R}^d \mapsto\{0,1,2\}$, let $$ R(f):=\mathbb{E}_{(X, Y) \sim \rho}\left[\mathbb{1}_{f(X) \neq Y}\right] $$ $f_B$ the Bayes classifier. can we get a family of ...
fantacy_crs's user avatar
3 votes
0 answers
63 views

How to prove emprical risk converges to expectation risk as $n\to \infty$?

For example, for a classical binary classification: $x \in \mathbb{R}^d$ and $y \in\{0,1\}$ let empirical risk be $R_{\ell}^n(f):=\frac{1}{n} \sum_{i=1}^n \ell\left(f\left(X_i\right), Y_i\right)$ and ...
fantacy_crs's user avatar
31 votes
5 answers
2k views

On average, how many uniformly random real numbers $u$ are needed for their sum to exceed $1$, if $u_1$ is in $(0,1)$ and $u_k$ is in $(0,eu_{k-1})$?

A well-known question is: on average, how many uniformly random real numbers in $(0,1)$ are needed for their sum to exceed $1$? The answer is $e$. Let's tweak this question by making each random ...
Dan's user avatar
  • 3,507
5 votes
1 answer
738 views

Maximum distance from origin of simple random walk

Let $\epsilon_1, \dots, \epsilon_n$ be random signs, equiprobably in $\{-1, 1\}$, independently. Let $S_k = \sum_{j=1}^k \epsilon_j$. I am wondering what is known about the expectation $$ \mathbb{E}\...
Drew Brady's user avatar
1 vote
0 answers
168 views

Expectation of inverse of complex Gaussian variables

If we consider a complex Gaussian random variable as $h\sim\mathcal{CN}(0,\gamma)$, where $\gamma$ is the variance. Is there any closed-form solution with $\gamma$ for $\mathbf{E}\left[\frac{1}{\lVert ...
Charlie Nie's user avatar
1 vote
1 answer
198 views

Probability of multivariant gaussian random variables in different areas

$\newcommand{\sgn}{\operatorname{sgn}}$Let $X_i$ is a gaussian random variable correlated with others. we want to find the probability of each possible case to find the expectation of following ...
A. R.'s user avatar
  • 25
0 votes
0 answers
171 views

How many elements have a "small" order in a finite field?

I'm hoping that this is an easy question for someone. How many elements can we expect to have multiplicative order at most $n^{1/c}$ in one of the finite fields $\mathbb{F}_p$ with $p$ prime with $n \...
Matt Groff's user avatar