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
1
vote
1
answer
130
views
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. ...
- 63
1
vote
0
answers
82
views
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'...
- 6,214
0
votes
0
answers
153
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 ...
- 63
0
votes
0
answers
32
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 ...
- 1
0
votes
1
answer
154
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 ...
- 103
0
votes
0
answers
39
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 ...
- 2,183
7
votes
1
answer
556
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:=\...
- 128k
0
votes
1
answer
122
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 $i=1,\ldots,k,$ $Y_i = \begin{cases} 1 & \text{if } X=a_i, \\ 0 & \text{otherwise.} \end{cases}$
\...
3
votes
0
answers
133
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 ...
- 153
2
votes
0
answers
87
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 ...
- 131
4
votes
1
answer
188
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 $...
- 53
0
votes
1
answer
108
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]{\...
- 177
2
votes
1
answer
212
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 ...
- 21
1
vote
1
answer
119
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?
$$\lim_{h\...
- 105
0
votes
1
answer
77
views
Estimation on rotationally-disturbed random vectors
During developing a new statistical estimator, I faced the following problem.
Let $\mathbf{x}_i$ be a sequence of i.i.d. $d$-dimensional random vectors with
\begin{align*}
\mathbf{x}_i = \mathbf{O}...
- 284
2
votes
0
answers
69
views
Rates of convergence in the functional CLT/weak invariance principle for martingale triangular arrays
There are results for the rate of convergence of the functional CLT/weak invariance principle for martingales difference sequences, for example theorem 4.5 in the book Martingale theory and its ...
- 301
-3
votes
1
answer
154
views
Proving that $P($$\{\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 ...
- 147
1
vote
0
answers
208
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 ...
- 21
3
votes
2
answers
428
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 ...
- 131
1
vote
0
answers
56
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
$$
\begin{align}
I_{n, l} &= \{ (l - 1)b_{n} + 1 + r_{n}, \ldots, lb_{n} \}, \quad l = ...
- 299
1
vote
1
answer
232
views
Hypothesis to guarantee Lindeberg's condition
Imagine to have a set of random variables $\{ X_i \}_{i=1}^{n}$ independent (Non identically distributed). In these scenario, if the Lindeberg's condition hold we can extend the result of the CLT, i.e....
- 305
-1
votes
1
answer
172
views
The distribution of the sum of a non-zero vector with random signs
Given a non-zero high-dimensional vector, $v\in (\mathbb{R} \setminus \{0\}) ^ d$, and a random sign vector $s \in \{-1,1\}^d$ (i.e., each entry is a rademacher random variable).
Empirically, I find ...
- 121
3
votes
1
answer
396
views
A convergence problem
I have a math/stat problem where I need to show the convergence of the average of a sequence of experiments to an interval. The sequence of experiments is not i.i.d., hence the standard law of large ...
- 108
3
votes
1
answer
436
views
Is the limit of compound Poisson random variables a compound Poisson r.v.?
Let $Y$ be an infinitely divisible (I.D.) random variable.
Let $\nu$ be any measure not necessarily finite: $\nu(\mathbb R)\leq \infty$. Suppose that $Y \sim (0, \nu,0)_0$ according to the notation on ...
- 13
5
votes
1
answer
288
views
Maximal inequality of iid random variables $\{X_{ij}\}_{1\leqslant i,j \leqslant n}$
Suppose that $\{X_{ij}\}_{1\leqslant i,j\leqslant n}$ are iid random variables with $\mathbb{E}(X_{11})=0$ and $\mathrm{Var}(X_{11})=1$, does the following convergence hold:
$$
\max_{1\leqslant j\...
- 81
4
votes
2
answers
348
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 ...
1
vote
0
answers
157
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
$$\|...
- 91
2
votes
1
answer
415
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 ...
- 23
3
votes
1
answer
76
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$. ...
- 193
1
vote
1
answer
88
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 ...
- 203
3
votes
1
answer
503
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 ...
- 433
4
votes
0
answers
203
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 (...
- 37.7k
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 ...
- 21
1
vote
0
answers
176
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. ...
- 11
6
votes
1
answer
169
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$,...
- 6,214
2
votes
1
answer
97
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/...
- 219
7
votes
1
answer
624
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 \...
- 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 ...
- 249
2
votes
1
answer
362
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 $$\frac{1}{\sqrt{n}}\sum_{k=1}^n(X_k-...
- 249
0
votes
0
answers
69
views
Law of large numbers over each mean of $h$ consecutive variables
Let $X_1, X_2, \dots$ be i.i.d. random variables with finite mean $\mu$.
The (weak) law of large numbers says that
$$\forall\varepsilon > 0\quad \lim_{n \to \infty} \mathbf{Pr}\!\left[\,\left|\...
- 11
0
votes
1
answer
649
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 ...
- 380
1
vote
1
answer
193
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 ...
- 6,853
4
votes
1
answer
363
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 $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 380
5
votes
0
answers
183
views
Central limit theorem versus entropy in dynamical systems context
A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
- 553
4
votes
0
answers
235
views
Convergence of the expectation of a random variable when conditioned on its sum with another, independent but not identically distributed
Suppose that for all $n \in \mathbf{N}$, $X_n$ and $Y_n$ are independent random variables with
$$X_n \sim \mathtt{Binomial}(n,1-q),$$
and
$$Y_n \sim \mathtt{Poisson}(n(q+\epsilon_n)),$$
where $q \in (...
- 91
0
votes
0
answers
122
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}&...
- 101
4
votes
1
answer
174
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
votes
1
answer
431
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 ...
- 2,239
1
vote
1
answer
273
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 ...
- 51
1
vote
1
answer
103
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....
