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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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Is there a proof of the de Moivre-Laplace central limit theorem along these lines?

Let $X_1, X_2, \dots$ denote independent identically distributed random variable with, say, distribution given by $P(X_i= \pm 1)=1/2$. As usual, set $$S_n=X_1+ \cdots +X_n.$$ It follows from Skorokhod'...
Keivan Karai's user avatar
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0 votes
0 answers
144 views

Inequalities on the distribution of the maximum of the normalized sum $\max_{k = 1,\dots,n} \frac{|S_k|}{\sqrt{k}}$

Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. random variables with $\mathbb{E}(X) = 0$,$\mathbb{E}(X^2) = \sigma^2$ and finite moments. Let $S_k = \sum_{i = 1}^{k} X_i$ and consider the normalized ...
MathRevenge's user avatar
0 votes
0 answers
31 views

The relay use of invariant set theory

For a dynamical system, set $A$ is an invariant set with a function $V_1$, whose derivative is semi negative definite on $A$, and the region where the derivative is $0$ is the set $B$, which is also ...
ya g's user avatar
  • 1
0 votes
1 answer
127 views

Limit distribution of the self-normalized sum of Cauchy random variables

This is something that has come up in my research. I originally posted this question on CrossValidated but realized it might be better suited for this site. I have deleted the question there (in case ...
FileHandler's user avatar
0 votes
0 answers
35 views

Multivariable local CLT for uncorrelated (but dependent) coordinates?

Let $\vec f, \vec g\sim\mathcal{N}(0, \sigma^2I_n)$ be independent Gaussians. Define $\mathsf{cyc}^i(\vec f) = (\vec f_i, \vec f_{i+1},\dots, \vec f_{n-1}, \vec f_0, \vec f_1,\dots, \vec f_{i-1})$ to ...
Mark Schultz-Wu's user avatar
7 votes
1 answer
541 views

A variation on the Borel–Cantelli lemma theme

Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let \begin{equation*} E:=\bigcap_{n\ge0}B_n, \end{equation*} where \begin{equation*} B_n:=\...
Iosif Pinelis's user avatar
0 votes
1 answer
119 views

Central limit theorems involving nominal-valued random variables

Suppose $X$ is a random variable taking values in a finite set $\{a_1,\ldots, a_k \}$ and for $i=1,\ldots,k,$ $Y_i = \begin{cases} 1 & \text{if } X=a_i, \\ 0 & \text{otherwise.} \end{cases}$ \...
Michael Hardy's user avatar
3 votes
0 answers
126 views

An analogue of Kolmogorov's law of the iterated logarithm

Let $X_1,\dots,X_n$ be independent random variables, each with mean zero and finite variance. Let $S_n = \sum\limits_{k=1}^n X_k$ and $s_n^2=ES_n^2$. We say the sequence obey the law of iterated ...
graham's user avatar
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2 votes
0 answers
84 views

Do Fagin's zero-one laws hold on stochastic block model?

Let $n$ be a positive integer (the number of vertices), $k$ be a positive integer (the number of communities), $p = (p_1, . . . , p_k)$ be a probability vector on $[k] := \{1, . . . , k\}$ (the prior ...
SagarM's user avatar
  • 131
4 votes
1 answer
183 views

Sign of error in the central limit theorem

Let $X_n$ and $Y_n$ be independent copies of two random variables $X$ and $Y$ with domain $\{-1,0,1\}$ for $n\in \mathbb{N}$. For a given $k\in \mathbb{N}$, I would like to find conditions on $X$ and $...
Flo Dorner's user avatar
0 votes
1 answer
106 views

Functional CLT with an asymptotically small time change

This question was posted to MathSE but it seems like MathOverflow might be the more appropriate place for it. Suppose I know that $(\frac{1}{\sqrt{m}}X(mt))_{0\leq t\leq 1}\xrightarrow[m\to\infty]{\...
user1598's user avatar
  • 177
2 votes
1 answer
183 views

How to show the joint weak convergence?

Given a $T>0$, let $\mathcal{C}[0,T]$ be the space of continuous functions on $[0,T]$. Let $Y_n(t)$ be stochastic processes in $\mathcal{C}[0,T]$. We define the weak convergence in the sense of ...
Mike HWANG's user avatar
1 vote
1 answer
117 views

calculating a double limit

We have the following term: $$ (e^{-a h}+e^{-b h})^n / 2^n$$ Now we take the limit: $$ h\to 0, n\to \infty $$ What relation of $h$ and $n$ must be satisfied for the following limit to hold? $$\lim_{h\...
Lili Si's user avatar
  • 105
0 votes
1 answer
74 views

Estimation on rotationally-disturbed random vectors

During developing a new statistical estimator, I faced the following problem. Let $\mathbf{x}_i$ be a sequence of i.i.d. $d$-dimensional random vectors with \begin{align*} \mathbf{x}_i = \mathbf{O}...
Seung Hyeon Yu's user avatar
2 votes
0 answers
60 views

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 ...
Aurelien's user avatar
  • 191
-3 votes
1 answer
151 views

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 ...
FD_bfa's user avatar
  • 147
1 vote
0 answers
200 views

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 ...
Charles's user avatar
  • 21
2 votes
2 answers
341 views

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 ...
Snoop's user avatar
  • 121
1 vote
0 answers
53 views

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 = ...
AlbertRapp's user avatar
1 vote
1 answer
185 views

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....
user1172131's user avatar
-1 votes
1 answer
146 views

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 ...
Amit Portnoy's user avatar
3 votes
1 answer
379 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 ...
Star's user avatar
  • 76
3 votes
1 answer
417 views

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 ...
PSE's user avatar
  • 13
5 votes
1 answer
275 views

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\...
MHMH's user avatar
  • 81
4 votes
2 answers
327 views

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 ...
Adrian Leverkuhn's user avatar
1 vote
0 answers
139 views

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 $$\|...
Hamid Enki's user avatar
2 votes
1 answer
391 views

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 ...
James Farre's user avatar
3 votes
1 answer
65 views

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 $p \to \infty$. ...
Olius's user avatar
  • 173
1 vote
1 answer
87 views

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 ...
Max's user avatar
  • 203
3 votes
1 answer
455 views

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 ...
CWC's user avatar
  • 433
4 votes
0 answers
186 views

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 (...
Brendan McKay's user avatar
2 votes
1 answer
2k 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 ...
sar1729's user avatar
  • 21
1 vote
0 answers
170 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. ...
Bert's user avatar
  • 11
6 votes
1 answer
154 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 $0$,...
Keivan Karai's user avatar
  • 6,162
2 votes
1 answer
93 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/...
Aftermath 12345's user avatar
7 votes
1 answer
549 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 \...
Dor's user avatar
  • 723
2 votes
1 answer
104 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 ...
Kurt.W.X's user avatar
  • 249
2 votes
1 answer
353 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-...
Kurt.W.X's user avatar
  • 249
0 votes
0 answers
66 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|\...
JerryAZ's user avatar
  • 11
0 votes
1 answer
583 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 ...
Jiyuan Zhang's user avatar
1 vote
1 answer
180 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 ...
dohmatob's user avatar
  • 6,824
4 votes
1 answer
349 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,\...
Jiyuan Zhang's user avatar
5 votes
0 answers
177 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 ...
jason's user avatar
  • 553
4 votes
0 answers
228 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 (...
as1's user avatar
  • 91
0 votes
0 answers
114 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
4 votes
1 answer
173 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, ...
vassilis papanicolaou's user avatar
4 votes
1 answer
422 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 ...
Hans's user avatar
  • 2,229
1 vote
1 answer
269 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 ...
Murali's user avatar
  • 51
1 vote
1 answer
99 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....
Jorge I. González C.'s user avatar
11 votes
0 answers
222 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 ...
Uchiha's user avatar
  • 87