Questions tagged [limit-theorems]

For questions about limit theorems of probability theory: (functional or not) central limit theorem, law of large numbers, law of iterated logarithm, etc.

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60 views

Finding a sequence from weak convergence

Let $(X_n)_n$ be a sequence of independent random variable, $(u_n)_n$ a sequence of positive numbers, such that $$\frac{1}{u_n}\sum_{k=1}^nX_k \Rightarrow X$$ where $X$ is not degenerate. Prove that ...
2
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1answer
69 views

Converse for the central limit theorem of $q$-dependent random variables

Let $(X_n)_n$ be a sequence of $q$-dependent random variables and identically distributed. If $E[X_1^2]<+\infty,$ then the Hoeffding-Robbins theorem states that $$\frac{1}{\sqrt{n}}\sum_{k=1}^n(X_k-...
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0answers
23 views

High-order associated Legendre polynomials

In a similar spirit (yet not the same) as in the question posted here, I am interested in finding the asymptotic expression for $$P_{a+ib}^{-c}(x)$$ for $c\sim b\gg a$ and for finite $x$. I would be ...
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0answers
56 views

Law of large numbers over each mean of $h$ consecutive variables

Let $X_1, X_2, \dots$ be i.i.d. random variables with finite mean $\mu$. The (weak) law of large numbers says that $$\forall\varepsilon > 0\quad \lim_{n \to \infty} \mathbf{Pr}\!\left[\,\left|\...
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1answer
74 views

Stable law and the domains of attraction

The multivariate generalised central limit theorem with their domains of attraction was given by Rvačeva (see also this post). The original paper is not very accessible on the internet, and neither ...
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0answers
40 views

Central limit theorem behavior in a mean-field model with quenched randomness

Let $\mathbf{h} := \{h_i \}_{i=1}^\infty$ be a sequence of i.i.d random variables and $J > 0$. For $n \in \mathbb{N}$ and a realization of $\mathbf{h}$, we define the Hamiltonian $H_n (\cdot, \...
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1answer
61 views

Compute limit of $\mathbb P(Y \le X_n)$ using limiting information on the sequence of random variables $X_n$

Let $Y$ be a symmetric random variable, $(X_n)_n$ be a sequence of nonnegative random variables, and set $p_n = \mathbb P(Y \le X_n)$. It is known from Slutsky's theorem that, if $c$ is a constant ...
2
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1answer
62 views

Reference for multivariate generalised CLT

I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$, $$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
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0answers
129 views

Central limit theorem versus entropy in dynamical systems context

A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem. Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
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0answers
102 views

Convergence of the expectation of a random variable when conditioned on its sum with another, independent but not identically distributed

Suppose that for all $n \in \mathbf{N}$, $X_n$ and $Y_n$ are independent random variables with $$X_n \sim \mathtt{Binomial}(n,1-q),$$ and $$Y_n \sim \mathtt{Poisson}(n(q+\epsilon_n)),$$ where $q \in (...
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0answers
42 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}&...
4
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1answer
134 views

A randomized central limit theorem

Let $X_k$, $k = 1, 2, \dots$, be a sequence of i.i.d. random variables with finite second moments. Also, let $N_k \geq 1$, $k = 1, 2, \dots$, be a sequence of random variables taking integral values, ...
4
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1answer
273 views

Central limit theorem for resampling

This is a cross-post from stats.stackexchange.com. No answer has appeared there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it. What is the analog ...
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1answer
203 views

Expected value of square[X/sigmaX] = 1/n^2(1+1/pi)?

Please see the below link for the complete description. I already have an answer shown in the link, based on many Excel simulations ($n=4$ to $100$, $x_i$ generated by RAND() function of Excel). I ...
1
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1answer
68 views

Dependent random variables converging to a density in mean

Let $X$ be an absolutely continuous r.v. with density $f$ which is continuous on $(0,\infty)$. Fix $x>0$ and consider some a.s. decreasing sequence $Y_n$ bounded by $X$ such that $Y_n\searrow 0$ a....
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168 views

Functional Weak Convergence of Maximum Likelihood Estimator

Let $\hat{\theta}_n$ be the Maximum Likelihood Estimator of parameter $\theta$, where $n$ is the sample size. It is well-known that under sufficient regularity conditions, we have the asymptotic ...
4
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1answer
148 views

Local central limit theorem far from the center

Let $X_i$ be a sequence of iid random variables, $E [X] = 0$, $E [X^2] = 1$ and $E [|X|^k] < \infty$ for some $k \ge 3$. Classical local CLT says that the density function $f_n$ of $\frac1{\sqrt n}...
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1answer
598 views

Large-n limit of the distribution of the normalized sum of Cauchy random variables

What is the large-n limit of a distribution of the following sample statistic:$$x\equiv\displaystyle\frac{\sum^{n}X_{i}}{\,\sqrt{\,\sum^{n}X_{i}^{2}\,}\,}$$ when sampling the Cauchy(0,1) distribution? ...
5
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1answer
473 views

Variance of sum of $m$ dependent random variables

I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here. Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random ...
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2answers
850 views

Can we do better than Azuma-Hoeffding when the variance is small?

The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
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0answers
109 views

Reference Request: Local Central Limit Type Theorem (CLT) for the Cycle

I am looking for a reference for a local CLT for the usual SRW on the cycle -- in continuous-time, ideally. I know the statement for a SRW (and a reference, say Lawler and Limic; Random Walk: A Modern ...
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0answers
147 views

Random $\beta$-transformation and its limit theorem

given probability space $ (\Omega, T, \mu), \mu$ is ergodic and $ T $ is invertible ( can regard $T$ as two sides shift) define random $\beta$-transformations: random variable $\beta:\Omega \to (1,\...
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1answer
156 views

Iterated logarithm and gaussian concentration : a paradox

Let $G_1, \dots, G_n$ be iid random variables, such that $G_1 \sim \mathcal N(0,1)$ Let $$S_n = \sum_{i=1}^n G_i\quad \text{and} \quad\tilde{S}_n = \frac{1}{\sqrt{2n\log\log n}}S_n$$ It is easy to ...
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0answers
69 views

“Optimal” local limit theorems for densities vanishing at zero

Consider a nonnegative stable distribution with a density that vanishes at zero, such as $$f(t)=\frac{e^{-1/2t}}{\sqrt{2\pi t^3}},\qquad t\geq0.$$ Suppose (for simplicity) that we have i.i.d copies $(...
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0answers
529 views

Calculating Wasserstein's distance between an empirical distribution and a combination of normal distributions

Context of the problem Let $\xi$ be a random variable (with real value) with support $\Xi=\mathbb{R}$ and $\xi_{1},\ldots,\xi_{N}$ be a sample of $\xi$. We consider the empirical probability $$\...
2
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1answer
277 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 ...
3
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1answer
90 views

A $t$-test for ordered pairs

Suppose I have random variables $$ W_i = \begin{cases} w_1 &\text{with prob. } p_1, \\ w_2 &\text{with prob. } p_2, \\ w &\text{with prob. } 1-p_1-p_2,\end{cases} \qquad i = 1, \dots, 2n+1....
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0answers
380 views

How to obtain the probability distribution of a sum of dependent discrete random variables more efficiently

I hope you are well. Here is my problem. Let $\{s_0,\,s_1,\ldots,\,s_T\}$ be a sequence of discrete random variables and denote $S_t=s_0+s_1+\cdots+s_t$, with $S_0=0$ and $S_T\leq M$, where $M$ and $T$...
3
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1answer
90 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 ...
3
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1answer
223 views

Approximating John's ellipsoid from uniform sampling of a centrally symmetric convex polyhedron

A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called ...
2
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2answers
505 views

Monotone convergence theorem for stochastic integrals

I was wondering if there exists an equivalent of a monotone convergence theorem for stochastic integrals. I looked into plenty of books and papers, but I haven't found anything useful. I would expect ...
4
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0answers
108 views

Concentration Inequality for Score Functions of Exponential Familty

Let $p$ be the density of a continuous one-parameter exponential family distribution on $\mathbb{R}$. We assume that $$p(x) = c(x)\cdot \exp\bigl [ x \cdot \theta - b(\theta ) \bigr ], $$ where $\...
3
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3answers
268 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}{...
5
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0answers
1k views

Skorokhod' representation theorem: What is (are) possible filtration(s) on the common probability space?

I asked this question on math.stackexchange at https://math.stackexchange.com/questions/1941142/skorokhods-representation-theorem-what-is-the-filtration-on-the-common-probabi and haven't received ...
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0answers
586 views

Product of two random Gaussian matrices - orthant probability

Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal ...
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0answers
141 views

Limit of stochastic subsequence of stationary ergodic sequence

Let $\{X_k\}_{k\in\mathbb{N}}$ be a stationary ergodic sequence on a probability space $(\Omega,\mathcal{F},P)$ with shift $T$. Also, let $\{v_k\}_{k\in\mathbb{N}}$ be a sequence of random variables ...
2
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1answer
281 views

Version of Donsker-Invariance-Principle

Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$...
13
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2answers
679 views

Is there a rate of convergence for Donsker's theorem?

For the standard CLT, one can easily estimate a rate of convergence if you assume that the random variables have a little more than two moments. Let $S_n$ be the centered-scaled sum of $n$ iid ...
3
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0answers
123 views

“Local” functional central limit theorem for the empirical distribution function

This question is a repost from Mathematics Stack Exchange, where it did not receive any answer. Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[...
9
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1answer
1k views

Law of large numbers for martingales

I apologize in advance if this question is too basic, but I've received no response on Math Stack Exchange, so perhaps it is more appropriate here: Let $X_n$ be a square-integrable martingale with $\...
13
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1answer
578 views

Applications of the Central Limit Theorem in dynamical systems

There are very many papers in the area of (possibly non-uniformly) hyperbolic dynamical systems whose aim is to prove the Central Limit Theorem. In a dynamical context, this means that one: has a ...
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0answers
73 views

CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...
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0answers
213 views

Almost independent Bernoulli variables

There is some global parameter $n\to\infty$. And a function $N=N(n)\to\infty$. Let $X^n_1,X^n_2,\ldots,X^n_N$ be independent Bernoulli random variables, where $\delta\le P(X^n_i=1)=1-P(X^n_i=0)\le 1-\...
4
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0answers
644 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 ...
2
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0answers
175 views

Strong law of large number for semimartingale

I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion. Thanks
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2answers
254 views

Reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws. Let $...
3
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0answers
86 views

Functions whose Laplace transforms have prescribed behavior at minus infinity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non-negative function with entire Laplace transform $\hat{f}$ (in particular $\lim_{t\to \infty}e^{st}f(t)=0$ for all $s$), and $p_0$ a positive integer. ...
4
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0answers
319 views

Unusual generalization of the law of large numbers

I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
16
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4answers
5k views

Rate of convergence in the Law of Large Numbers

I'm working on a problem where I need information on the size of $E_n=|S_n-n\mu|$, where $S_n=X_1+\ldots+X_n$ is a sum of i.i.d. random variables and $\mu=\mathbb EX_1$. For this to make sense, the $(...
3
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1answer
230 views

Berry-Esseen bound in 2 dimensions for linear combinations

Let us say have a sequence of $n$ 2-$D$ random variables $X_i=(\varepsilon_i/\sqrt{n},i\varepsilon_{i}\sqrt{6}/n^{3/2})$, where $\varepsilon_{i}$ are independent random variables such that $\mathbb{P}(...