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.
109 questions
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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 ...
CommunityBot
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Breiman's first exit times from a square root boundary generalization
The paper "First exit times from a square root boundary" by Breiman, generalizes an observation made by Blackwell and Freedman. In summary: given a zero-mean random walk $S_n$ with i.i.d. ...
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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'...
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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 ...
CommunityBot
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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\...
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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 ...
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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 ...
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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 ...
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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:=\...
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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}$
\...
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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 ...
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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 ...
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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]{\...
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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 $...
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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 ...
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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\...
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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}...
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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 ...
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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 ...
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Proving that $P($$\{\text{$a$ and $b$ are co-prime}$ }$)=0$ for $a,b$ following the Uniform distribution over $[n, 2n]$ as $n \rightarrow \infty$
I have been working on a problem concerning the "line of sight" from a fixed integer co-ordinate — let's say $(0,0)$ — to a variable co-ordinate $(a,b)$. Having a line of sight means that ...
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CLT for dependant random variables
I define a distribution of probability $L$ on $C:=C_0([0,1],\mathbb{R})$ the set of continuous functions $f$ on $[0,1]$ such that $f(0)=f(1)=0$. I suppose that $L$ is centered and has a covariance ...
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Rate of convergence in the Law of Large Numbers
I'm working on a problem where I need information on the size of $E_n=|S_n-n\mu|$, where $S_n=X_1+\ldots+X_n$ is a sum of i.i.d. random variables and $\mu=\mathbb EX_1$. For this to make sense, the $(...
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A limit definition of regular conditional probability
I am working with a proof that would greatly benefit from a definition of conditional probability along the lines of the obscure unreferenced alternative definition found in Wikipedia. A Wikipedia ...
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Bounding difference of characteristic functions with mixing coefficients
Setting
Let $X = \{ X(t), t \in \mathbb{R} \}$ be a stationary, $\alpha$-mixing stochastic process and define
$$
\begin{align}
I_{n, l} &= \{ (l - 1)b_{n} + 1 + r_{n}, \ldots, lb_{n} \}, \quad l = ...
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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....
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$\|...
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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$. ...
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CLT for stationary sequences with infinite variance
There is a well-known central limit theorem for as a stationary sequences.
If $( X_n )_n$ is a stationary sequence and $E X_n=0$ then under suitable mixing conditions the sequence $S_n := n^{-1/2}\...
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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 ...
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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 (...
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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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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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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. ...
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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$,...
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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/...
CommunityBot
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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 \...
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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 ...
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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 $$\frac{1}{\sqrt{n}}\sum_{k=1}^n(X_k-...
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Stable law and the domains of attraction
The multivariate generalised central limit theorem with their domains of attraction was given by Rvačeva (see also this post). The original paper is not very accessible on the internet, and neither ...
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Law of large numbers over each mean of $h$ consecutive variables
Let $X_1, X_2, \dots$ be i.i.d. random variables with finite mean $\mu$.
The (weak) law of large numbers says that
$$\forall\varepsilon > 0\quad \lim_{n \to \infty} \mathbf{Pr}\!\left[\,\left|\...
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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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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 $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 128k
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Central limit theorem versus entropy in dynamical systems context
A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
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Convergence of the expectation of a random variable when conditioned on its sum with another, independent but not identically distributed
Suppose that for all $n \in \mathbf{N}$, $X_n$ and $Y_n$ are independent random variables with
$$X_n \sim \mathtt{Binomial}(n,1-q),$$
and
$$Y_n \sim \mathtt{Poisson}(n(q+\epsilon_n)),$$
where $q \in (...
CommunityBot
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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}&...
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1
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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, ...
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