# 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.

77
questions

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votes

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286 views

### Central limit theorem for weak correlated random variables

I have a sequence of weak correlated continuous random variables $\{X_i\}$ with bounded variance and $\operatorname{Cov}(X_i,X_j)\rightarrow0$ for $|i-j|\rightarrow\infty$.
I was able to find a ...

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**0**answers

149 views

### Functional version of a specific martingale central limit theorem

I am looking for a functional version of Theorem 1b of Heyde. The result states:
Theorem (Theorem 1b in Heyde): Suppose that $(M_n)_{n \geq 1}$ is a square-integrable martingale with mean zero. ...

**6**

votes

**1**answer

106 views

### Uniform upper bounds for the return probability of random walks on $ \mathbb{R}$

Let $\mu$ be a probability measure with finite support on integers or the real line with the property that $\mu( 0) \le p$ for a fixed $0<p <1$. Let $S_n$ denote the random walk starting at $0$,...

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vote

**1**answer

80 views

### Local limit theorems for circular/spherical distributions

Here are some of the classical density functions for spherical distributions (on the $\mathcal{S}^{d-1}$ sphere, living in the Euclidean space $\mathbb{R}^d$):
$$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/...

**7**

votes

**1**answer

238 views

### Local limit theorem for random walks on $\mathbb Z^d$

I'm looking for a reference for the following claim.
Let $W(n)$ be a centered random walk on $\mathbb Z^d$ with $W(0)=0$.
Suppose that $W(n)$ has a finite second moment.
Let $n\ge 1 $ and $k \in \...

**2**

votes

**1**answer

79 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**

votes

**1**answer

157 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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58 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|\...

**0**

votes

**1**answer

143 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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**1**answer

71 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 ...

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votes

**1**answer

100 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,\...

**5**

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**0**answers

151 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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118 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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votes

**0**answers

56 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}&...

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votes

**1**answer

142 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, ...

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**1**answer

323 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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vote

**1**answer

226 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 ...

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vote

**1**answer

74 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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187 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**

votes

**1**answer

152 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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votes

**1**answer

655 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? ...

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votes

**1**answer

637 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 ...

**12**

votes

**2**answers

1k 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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116 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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votes

**1**answer

174 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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71 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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**0**answers

654 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
$$\...

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votes

**1**answer

322 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 ...

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votes

**1**answer

91 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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418 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**

votes

**1**answer

96 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 ...

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votes

**1**answer

248 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 ...

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votes

**2**answers

611 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 ...

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**0**answers

118 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 $\...

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votes

**3**answers

277 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}{...

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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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**0**answers

613 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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**0**answers

150 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 ...

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votes

**1**answer

305 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 $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$...

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**3**answers

851 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 ...

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**0**answers

133 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[...

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votes

**1**answer

2k 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 $\...

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votes

**1**answer

637 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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**0**answers

77 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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220 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-\...

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**0**answers

776 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 ...

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**0**answers

199 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

**5**

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**2**answers

269 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 $...

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**0**answers

93 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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**0**answers

322 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 ...