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
votes
0
answers
40
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 ...
- 141
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 $u\in \mathbb{R}^n$,
$$
\mathbb{E}\left[e^{u^\top X_i}\right]\le e^{\frac{...
- 405
-4
votes
1
answer
141
views
Proving that $P($$\{\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 ...
- 137
1
vote
0
answers
113
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
1
vote
2
answers
92
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 ...
- 121
1
vote
0
answers
47
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
$$
\begin{align}
I_{n, l} &= \{ (l - 1)b_{n} + 1 + r_{n}, \ldots, lb_{n} \}, \quad l = ...
- 299
1
vote
1
answer
100
views
Hypothesis to guarantee Lindeberg's condition
Imagine to have a set of random variables $\{ X_i \}_{i=1}^{n}$ independent (Non identically distributed). In these scenario, if the Lindeberg's condition hold we can extend the result of the CLT, i.e....
- 121
-1
votes
1
answer
88
views
The distribution of the sum of a non-zero vector with random signs
Given a non-zero high-dimensional vector, $v\in (\mathbb{R} \setminus \{0\}) ^ d$, and a random sign vector $s \in \{-1,1\}^d$ (i.e., each entry is a rademacher random variable).
Empirically, I find ...
- 119
2
votes
1
answer
317
views
A convergence problem
I have a math/stat problem where I need to show the convergence of the average of a sequence of experiments to an interval. The sequence of experiments is not i.i.d., hence the standard law of large ...
- 57
3
votes
2
answers
261
views
Is the limit of compound Poisson random variables a compound Poisson r.v.?
Let $Y$ be an infinitely divisible (I.D.) random variable.
Let $\nu$ be any measure not necessarily finite: $\nu(\mathbb R)\leq \infty$. Suppose that $Y \sim (0, \nu,0)_0$ according to the notation on ...
- 217
5
votes
1
answer
182
views
Maximal inequality of iid random variables $\{X_{ij}\}_{1\leqslant i,j \leqslant n}$
Suppose that $\{X_{ij}\}_{1\leqslant i,j\leqslant n}$ are iid random variables with $\mathbb{E}(X_{11})=0$ and $\mathrm{Var}(X_{11})=1$, does the following convergence hold:
$$
\max_{1\leqslant j\...
- 71
4
votes
2
answers
213
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
72
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
217
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
48
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$. ...
- 43
1
vote
1
answer
82
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
2
votes
1
answer
288
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 ...
- 297
4
votes
0
answers
154
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 (...
- 35.6k
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 ...
- 21
1
vote
0
answers
159
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
126
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$,...
- 5,842
2
votes
1
answer
89
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
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 \...
- 723
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 ...
- 249
2
votes
1
answer
278
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 $$\frac{1}{\sqrt{n}}\sum_{k=1}^n(X_k-...
- 249
0
votes
0
answers
61
views
Law of large numbers over each mean of $h$ consecutive variables
Let $X_1, X_2, \dots$ be i.i.d. random variables with finite mean $\mu$.
The (weak) law of large numbers says that
$$\forall\varepsilon > 0\quad \lim_{n \to \infty} \mathbf{Pr}\!\left[\,\left|\...
- 11
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 ...
- 319
1
vote
1
answer
127
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 ...
- 6,338
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 $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 319
5
votes
0
answers
164
views
Central limit theorem versus entropy in dynamical systems context
A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
- 563
4
votes
0
answers
189
views
Convergence of the expectation of a random variable when conditioned on its sum with another, independent but not identically distributed
Suppose that for all $n \in \mathbf{N}$, $X_n$ and $Y_n$ are independent random variables with
$$X_n \sim \mathtt{Binomial}(n,1-q),$$
and
$$Y_n \sim \mathtt{Poisson}(n(q+\epsilon_n)),$$
where $q \in (...
- 91
0
votes
0
answers
87
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
159
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
375
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 ...
- 2,169
1
vote
1
answer
252
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
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
205
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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 ...
- 77
4
votes
1
answer
178
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}...
- 189
13
votes
1
answer
729
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}\,}\,}$$ when sampling the Cauchy(0,1) distribution? ...
- 409
5
votes
1
answer
821
views
Variance of sum of $m$ dependent random variables
I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here.
Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random ...
- 329
12
votes
2
answers
2k
views
Can we do better than Azuma-Hoeffding when the variance is small?
The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
- 1,721
3
votes
0
answers
127
views
Reference Request: Local Central Limit Type Theorem (CLT) for the Cycle
I am looking for a reference for a local CLT for the usual SRW on the cycle -- in continuous-time, ideally.
I know the statement for a SRW (and a reference, say Lawler and Limic; Random Walk: A Modern ...
- 540
2
votes
1
answer
197
views
Iterated logarithm and gaussian concentration : a paradox
Let $G_1, \dots, G_n$ be iid random variables, such that $G_1 \sim \mathcal N(0,1)$
Let $$S_n = \sum_{i=1}^n G_i\quad \text{and} \quad\tilde{S}_n = \frac{1}{\sqrt{2n\log\log n}}S_n$$
It is easy to ...
- 235
2
votes
0
answers
75
views
"Optimal" local limit theorems for densities vanishing at zero
Consider a nonnegative stable distribution with a density that vanishes at zero, such as
$$f(t)=\frac{e^{-1/2t}}{\sqrt{2\pi t^3}},\qquad t\geq0.$$
Suppose (for simplicity) that we have i.i.d copies $(...
- 851
2
votes
0
answers
711
views
Calculating Wasserstein's distance between an empirical distribution and a combination of normal distributions
Context of the problem
Let $\xi$ be a random variable (with real value) with support $\Xi=\mathbb{R}$ and $\xi_{1},\ldots,\xi_{N}$ be a sample of $\xi$. We consider the empirical probability
$$\...
- 159
2
votes
1
answer
440
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 ...
- 21
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....
- 267
5
votes
0
answers
450
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 $T$...
- 173
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 ...
- 4,881
3
votes
1
answer
267
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 ...
- 13.2k