# 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.

95
questions

2
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0
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40
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### Rates of convergence in the functional CLT/weak invariance principle for martingale triangular arrays

There are results for the rate of convergence of the functional CLT/weak invariance principle for martingales difference sequences, for example theorem 4.5 in the book Martingale theory and its ...

0
votes

0
answers

25
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### concentration inequality with matrix coefficient

Let $(X_i)_{i=1}^N$ be mean zero sub-Gaussian random vectors in $\mathbb{R}^n$, i.e., there exists $C>0$ such that for all $u\in \mathbb{R}^n$,
$$
\mathbb{E}\left[e^{u^\top X_i}\right]\le e^{\frac{...

-4
votes

1
answer

141
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### Proving that $P($$\{\text{$a$ and $b$ are co-prime}$ }$)=0$ for $a,b$ following the Uniform distribution over $[n, 2n]$ as $n \rightarrow \infty$

I have been working on a problem concerning the "line of sight" from a fixed integer co-ordinate — let's say $(0,0)$ — to a variable co-ordinate $(a,b)$. Having a line of sight means that ...

1
vote

0
answers

113
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### CLT for dependant random variables

I define a distribution of probability $L$ on $C:=C_0([0,1],\mathbb{R})$ the set of continuous functions $f$ on $[0,1]$ such that $f(0)=f(1)=0$. I suppose that $L$ is centered and has a covariance ...

1
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2
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92
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### A limit definition of regular conditional probability

I am working with a proof that would greatly benefit from a definition of conditional probability along the lines of the obscure unreferenced alternative definition found in Wikipedia. A Wikipedia ...

1
vote

0
answers

47
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### Bounding difference of characteristic functions with mixing coefficients

Setting
Let $X = \{ X(t), t \in \mathbb{R} \}$ be a stationary, $\alpha$-mixing stochastic process and define
$$
\begin{align}
I_{n, l} &= \{ (l - 1)b_{n} + 1 + r_{n}, \ldots, lb_{n} \}, \quad l = ...

1
vote

1
answer

100
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### Hypothesis to guarantee Lindeberg's condition

Imagine to have a set of random variables $\{ X_i \}_{i=1}^{n}$ independent (Non identically distributed). In these scenario, if the Lindeberg's condition hold we can extend the result of the CLT, i.e....

-1
votes

1
answer

88
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### The distribution of the sum of a non-zero vector with random signs

Given a non-zero high-dimensional vector, $v\in (\mathbb{R} \setminus \{0\}) ^ d$, and a random sign vector $s \in \{-1,1\}^d$ (i.e., each entry is a rademacher random variable).
Empirically, I find ...

2
votes

1
answer

317
views

### A convergence problem

I have a math/stat problem where I need to show the convergence of the average of a sequence of experiments to an interval. The sequence of experiments is not i.i.d., hence the standard law of large ...

3
votes

2
answers

261
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### Is the limit of compound Poisson random variables a compound Poisson r.v.?

Let $Y$ be an infinitely divisible (I.D.) random variable.
Let $\nu$ be any measure not necessarily finite: $\nu(\mathbb R)\leq \infty$. Suppose that $Y \sim (0, \nu,0)_0$ according to the notation on ...

5
votes

1
answer

182
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### Maximal inequality of iid random variables $\{X_{ij}\}_{1\leqslant i,j \leqslant n}$

Suppose that $\{X_{ij}\}_{1\leqslant i,j\leqslant n}$ are iid random variables with $\mathbb{E}(X_{11})=0$ and $\mathrm{Var}(X_{11})=1$, does the following convergence hold:
$$
\max_{1\leqslant j\...

4
votes

2
answers

213
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### Does the average of correlated Gaussian random variables with mean zero and different variances converge in probability to their mean?

Let $X_i\sim N(0,\sigma_i^2)$ and $\operatorname{Corr}(X_i,X_j)>0$. Is it possible to show that $$\frac{1}{N} \sum_{i=1}^N X_i \overset{p}\rightarrow E[X_i]=0.$$ Do you have a reference to a law of ...

1
vote

0
answers

72
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### Motivation for Ionescu Tulcea-Marinescu (Lasota-Yorke inequality)

I wonder about motivations of a work of Ionescu Tulcea-Marinescu. In order to establish the decomposition of the operator $T$ they assume (condition (1.3)) this operator satisfies the inequality
$$\|...

2
votes

1
answer

217
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### Self normalized sum of products of i.i.d. random variables

Let $p\in (0,1)$ and $X_1, X_2, ...X_n \sim \text{Bern}(p)$ be $n$ i.i.d. Bernoulli random variables, where the probability that $X_i$ is $1$ equals $p$.
Fix $a,b>0$ different from $1$ that satisfy ...

3
votes

1
answer

48
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### When do Orlicz norms tend to the uniform norm?

It is well known that the $p$-norms tend to the $\infty$-norm, in that if $\lVert f \rVert_q < \infty$ for some $q \ge 1$ then $\lVert f \rVert_p \to \lVert f \rVert_\infty$ as $p \to \infty$. ...

1
vote

1
answer

82
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### Stationary and limiting distributions

Consider a CT Markov Process $X=(X_t)_{t\geq0}$ with state space $E\in\mathbb{R}^N$. Are there any general conditions under which a stationary distribution $\pi$ for $X$ is also a limiting ...

2
votes

1
answer

288
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### Weakly dependent central limit theorem

Say I have $N$ random variables $X_1,\cdots,X_i,\cdots,X_N$, with zero mean and finite variance. $X_i$ and $X_j$ are independent iif $|i-j|>m$, and positively correlated otherwise (say the ...

4
votes

0
answers

154
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### log-concavity and local CLT

If a sequence of 1-dimensional log-concave integer-valued distributions satisfies a Central Limit Theorem (CLT) and has variance going to $\infty$, then it satisfies a Local Central Limit Theorem (...

2
votes

1
answer

1k
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### 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 ...

1
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0
answers

159
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### 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

126
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### 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 $0$,...

2
votes

1
answer

89
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### 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

379
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

92
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

278
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### 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-...

0
votes

0
answers

61
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

333
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 ...

1
vote

1
answer

127
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### 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 ...

3
votes

1
answer

245
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### 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
votes

0
answers

164
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### 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 ...

4
votes

0
answers

189
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 (...

0
votes

0
answers

87
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### 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
votes

1
answer

159
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### 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
votes

1
answer

375
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### 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 ...

1
vote

1
answer

252
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### 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
vote

1
answer

83
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....

10
votes

0
answers

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### 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

178
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### 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}...

13
votes

1
answer

729
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### 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
votes

1
answer

821
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### 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

2k
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### 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 $\|...

3
votes

0
answers

127
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### 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 ...

2
votes

1
answer

197
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### 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 ...

2
votes

0
answers

75
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### "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 $(...

2
votes

0
answers

711
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
votes

1
answer

440
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### 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
votes

1
answer

100
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....

5
votes

0
answers

450
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### 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

104
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
votes

1
answer

267
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### 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 ...