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Finding a connection between two types of convergence

Please, help me find connections between two types of convergence: Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences: 1) $X_n \...
Ivan Petrov's user avatar
3 votes
1 answer
829 views

The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$

In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
user avatar
3 votes
1 answer
400 views

Countable convergence-determining class for weak convergence of probability measures

Suppose that $E$ is a Polish space. Portmanteau theorem asserts that a sequence $(\mu_n)$ of Borel probability measures weakly converges to a Borel probability measure $\mu$ (shortly, $\mu_n\overset{...
user154110's user avatar
2 votes
1 answer
99 views

Convergence of estimator given by a fixed point

Let $X$ be a non-negative random variable with cdf $F$ and define $$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function. Let $s_0$ be the unique fixed point of $G$. Now let $X_1,\dots,X_t$ ...
T. Tharoor's user avatar
2 votes
0 answers
122 views

Convergence of Bayesian posterior

Let $\Delta [0,1]$ denote the set of all probability distributions on the unit interval. Let $\mu \in \Delta [0,1]$ denote an arbitrary prior. Importantly, $\mu$ does not necessarily admit a density ...
user_newbie10's user avatar
0 votes
0 answers
123 views

Tightness of a uniformly bounded sequence of functions integrated with respect to a semimartingale

I am reading this paper by Jacod, Jakubowski and Mémin. In the proof of Theorem 1.3 the authors define, for each $n\geq1$ the function $\phi_n$ by $\quad\phi_n(s)=i+1-ns,\quad\text{if } \frac{i}{n}&...
jakobdt's user avatar
  • 101
6 votes
0 answers
120 views

Intuition behind the local limit theorem in hyperbolic groups

Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Denote by $X_n$ the induced random walk. Finally, let $p_n=\mu^{*n}(e)=P_e(X_n=e)$. The local limit ...
M. Dus's user avatar
  • 2,090
4 votes
1 answer
478 views

Order statistic - Rate of convergence of a p-quantile to the expectation

Fix some $k\in\mathbb N$ and some probability $p\in[0,1]$. Denote with $F_n$ the cdf of the k-th highest oder statistic (i.e. the distribution of the k-th highest draw) of $n$ draws from a uniform ...
jonasvw's user avatar
  • 43
4 votes
2 answers
145 views

Understanding equiprobable trinomial identity

With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
maliesen's user avatar
  • 284
2 votes
0 answers
64 views

Convergence of gPC expansions for random variables in the total variation distance

Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider ...
user avatar
2 votes
1 answer
185 views

Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$

Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s. ...
dohmatob's user avatar
  • 6,853
7 votes
1 answer
261 views

Comparison of several topologies for probability measures

Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
Kass's user avatar
  • 243
2 votes
0 answers
100 views

Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes

Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
Sam OT's user avatar
  • 560
2 votes
1 answer
254 views

Asymptotic rate for the expected value of the square root of sample average

I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$. Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$. I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$. My first ...
Florian Tramèr's user avatar
1 vote
2 answers
287 views

Showing $o(1)$ convergence for ratio of successive binomial tail probabilities

For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that $$ \frac{P(X>c)}{P(X>c-1)}=1-o(1) $$ uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...
stats134711's user avatar
5 votes
3 answers
5k views

Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
dohmatob's user avatar
  • 6,853
5 votes
0 answers
1k views

Asymptotic behavior of row sums in 2-d array of random variables

Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables: $B^m_{1,1}$ $B^...
cosmo-grant's user avatar
0 votes
0 answers
691 views

$L_1$ convergence for a product of indicator functions

Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions $$ \lim_{N\...
Marc's user avatar
  • 479
3 votes
1 answer
177 views

Convergence of SDEs

Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
ABIM's user avatar
  • 5,405
2 votes
2 answers
152 views

Divergence rate of geometric sum of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\...
Marc's user avatar
  • 479
3 votes
1 answer
473 views

Expected value of the maximum of the periodogram

Let us suppose that $X_1,\ldots,X_n$ with $n\ge1$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$ and define the DFT of $X_1,\ldots,...
Cm7F7Bb's user avatar
  • 423
1 vote
0 answers
87 views

Conditonal convergence implies convergence?

Note : All measures below are probability measures. Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$. Actually,...
Gericault's user avatar
  • 245
1 vote
1 answer
108 views

Maximum of the periodogram of a truncated sequence

Suppose that $Z,Z_1,Z_2,\ldots$ are iid random variables such that $\operatorname EZ=0$, $\operatorname EZ^2=1$ and $\operatorname E|Z|^s<\infty$ with some $s>2$. Let $\tilde Z_t=Z_tI_{\{|Z_t|\...
Cm7F7Bb's user avatar
  • 423
1 vote
1 answer
81 views

Convergence rate of $\operatorname E|\langle X,f_n\rangle|^p$

Suppose that $X$ is a random element with values in a separable Hilbert space $\mathbb H$ such that $\operatorname EX=0$ and $\operatorname E\|X\|^2<\infty$. Suppose that $f_1,f_2,\ldots$ form an ...
Cm7F7Bb's user avatar
  • 423
2 votes
0 answers
137 views

Rate of convergence of a test statistic towards a Gaussian random variable

This is a follow-up question to Rate of convergence of $\frac{1}{\sqrt{n\ln n}}(\sum_{k=1}^n 1/\sqrt{X_k}-2n)$, $X_i$ i.i.d. uniform on $[0,1]$? . My motivation is to construct a statistic whose rate ...
Przemek Repetowicz's user avatar
1 vote
0 answers
88 views

Questions about generalized Polynomial Chaos, book by Dongbin Xiu

I have some questions about Chapter 5 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu. Theorem 5.7: Let $Y$ be a random variable and $\mathbb{E}[Y^2]<\infty$. Let $Z$ ...
jum's user avatar
  • 31
27 votes
4 answers
3k views

Rate of convergence of $\frac{1}{\sqrt{n\ln n}}(\sum_{k=1}^n 1/\sqrt{X_k}-2n)$, $X_i$ i.i.d. uniform on $[0,1]$?

Let $(X_n)$ be a sequence of i.i.d. random variables uniformly distributed in $[0,1]$; and, for $n\geq 1$, set $$ S_n = \sum_{k=1}^n \frac{1}{\sqrt{X_k}}\,. $$ It follows from the generalized central ...
Clement C.'s user avatar
  • 1,372
1 vote
0 answers
202 views

Convergence in probability in the setting of free probability

Let $A_n$ and $B_n$ be sequences of positive-definite random matrices whose empirical spectral distributions converge to (possibly different) limiting spectral distributions $\mathcal A$ and $\mathcal ...
Apprentice's user avatar
4 votes
0 answers
2k views

Does rate of convergence in probability come from a metric?

In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
Froomfondel's user avatar
3 votes
1 answer
196 views

Uniform Convergence for Vectors

$\textbf{Problem statement:}$ Let $\mathcal H:\mathcal X \rightarrow \{0,1\}$ be a class of Boolean functions for $\mathcal X \subset \mathbb R^n$, and let the VC Dimension of $\mathcal H$ be $VC_{...
AvidLearner's user avatar
3 votes
1 answer
751 views

Wasserstein convergence of conditional measures

Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the same atoms): ...
JohnA's user avatar
  • 710
2 votes
1 answer
643 views

Weak convergence in Skorohod space

I am reading a paper where they prove Donsker's invariance principle for a sequence of dependent RV's. They do the following steps which I can't follow so well. I won't write out the precise ...
Nat's user avatar
  • 21
3 votes
1 answer
125 views

Convergence of function of stochastic processes

Let $X_t$ be a fixed cadlag semi-martingale and $J_n$ be a fixed sequence of functions from $\mathbb{R}^d$ to $\mathbb{R}$ which are twice continuously differentiable. If $J_n$ converge pointwise to ...
ABIM's user avatar
  • 5,405
5 votes
1 answer
295 views

Constructive Central Limit Theorem

Background: Let $\{a_i\}_{i=1}^n$ be i.i.d. random variables with zero-mean and unit variance, on a probability space $\Omega$. Define $$s_n=\frac{1}{\sqrt{n}}\sum_{i\leq n} a_i$$ Central limit ...
ecstasyofgold's user avatar
3 votes
1 answer
182 views

Superlinear Convergence of a Markov Chain

Suppose that we have a Markov process $\{Z_t\}_{t=0}^\infty$, where $Z_t \geq 0$ for any $t$. Assume that, conditioning on $Z_t = z_t$, we have $ \mathbb{E}\{Z_{t+1}|Z_t = z_t\} \leq \kappa z_t^2 $. ...
Minkov's user avatar
  • 1,127
4 votes
1 answer
265 views

Can a probability distribution from summing alternating signs have atoms?

Suppose $a_n > 0$ is a sequence of real numbers in $l^2 \setminus l^1$. i.e. $\sum a_n^2 < \infty$ but $\sum a_n = \infty$. If $B_n$ are an infinite sequence of independent Bernoulli random ...
David R. MacIver's user avatar
3 votes
3 answers
292 views

A question in central limit theorem

Suppose $\{X_n,n\ge1\}$ are independent r.v., $E(X_n)=0$, $\operatorname{Var} \left(X_n\right)=\sigma_n^2<\infty$. Set $S_n=\sum_{i=1}^nX_i$ and $s_n^2=\sum_{i=1}^n\sigma_i^2$, assume $$\frac{S_n}{...
J.Mike's user avatar
  • 141
2 votes
2 answers
736 views

Submartingales bounded in $L^p$, $p>1$

Let $p>1$ be a real number. It is known that if $(X_n)_{n\geq 0}$ is a martingale bounded in $L^p$ (i.e. $\sup\{\mathbb{E}(|X_n|^p), n\geq 0\} < +\infty$ ), then $(X_n)_{n\geq 0}$ converges a....
user avatar
3 votes
2 answers
1k views

Is there a notion of Convergence in PDF/PMF

I am learning about local limit theorems. The following example is probably why we don't have a "convergence in density/pmf." Ex: $X_1,X_2,\ldots$ is a sequence of independent RVs with mean $a$ and ...
The Substitute's user avatar
1 vote
1 answer
153 views

Rate of convergence of exponential of sample mean to exponential of first moment?

If $X_n \sim N(\mu,\sigma)$ and $T_n = \frac{1}{n}\sum_1^n X_i$ What is the rate of convergence of $e^{T_n}$ to $e^{\mu}$
user avatar
3 votes
2 answers
226 views

Liminf of the maximum of two iid sequences

Let $\{X_t\}_{t\ge1}$ and $\{Y_t\}_{t\ge1}$ be two iid sequences of random variables that have full support. That is, if $A\subseteq\mathbb{R}$ has positive Lebesgue measure, then $P(X\in A) >0$ ...
Marc's user avatar
  • 479
3 votes
1 answer
377 views

Law of large numbers for random functions?

Is there a version of the law of large numbers for random functions of the type: $h(X_j,\hat{\theta}_n)$, where $X_1,\dots,X_n$ are i.i.d. random variables, with distribution $F$, and $\hat{\theta}_n =...
Park's user avatar
  • 31
2 votes
0 answers
207 views

markov processes and ergodic theory

For an ergodic Markov Chain $$ \frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f] $$ where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
jkt's user avatar
  • 169
12 votes
1 answer
330 views

Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
user3605620's user avatar
8 votes
2 answers
537 views

Famous results about the value of a given limit assuming it exists

Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
Sylvain JULIEN's user avatar
9 votes
1 answer
556 views

Berry-Esseen bound for martingale sequence with varying and dependent variances

Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e. $$ E[X_{k}|\mathcal{F}_{k-1}] = 0 $$ where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$. Let $\sigma_{...
Nikolayevich's user avatar
3 votes
1 answer
105 views

How to show monotonocity and the limit? [closed]

Let me reformulate my recent question. Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density: $$\phi(x) = C\left\{ \begin{array}{lcc} \sqrt{...
smyroosh's user avatar
6 votes
0 answers
1k views

Interplay between CLT and convergence in Total Variation

Given a random variable $X$ with bounded moments such that $E[X] = 0, E[X^2] = 1$, let $F_n$ denote the distribution $\sum\limits_{i=1}^d\frac{X_i}{\sqrt{n}}$ where each $X_i$ is an independent copy ...
Cain's user avatar
  • 393
2 votes
1 answer
444 views

Literature question on the convergence rate of the empirical distribution

Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each $...
user avatar
4 votes
2 answers
864 views

Uniform Convergence of Moment Generating Function

In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as: $$ \begin{...
Rodrigo Zepeda's user avatar