All Questions
Tagged with limits-and-convergence pr.probability
184 questions
2
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0
answers
81
views
Convergence of random operators
I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
- 121
2
votes
0
answers
175
views
Representing a continuous time-inhomogeneous Markov chain by a stochastic integral
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
- 31
2
votes
0
answers
122
views
Convergence of Bayesian posterior
Let $\Delta [0,1]$ denote the set of all probability distributions on the unit interval.
Let $\mu \in \Delta [0,1]$ denote an arbitrary prior. Importantly, $\mu$ does not necessarily admit a density ...
- 131
2
votes
0
answers
64
views
Convergence of gPC expansions for random variables in the total variation distance
Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider ...
user85801
2
votes
0
answers
100
views
Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes
Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
- 560
2
votes
0
answers
137
views
Rate of convergence of a test statistic towards a Gaussian random variable
This is a follow-up question to Rate of convergence of $\frac{1}{\sqrt{n\ln n}}(\sum_{k=1}^n 1/\sqrt{X_k}-2n)$, $X_i$ i.i.d. uniform on $[0,1]$? . My motivation is to construct a statistic whose rate ...
2
votes
0
answers
207
views
markov processes and ergodic theory
For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
- 169
2
votes
0
answers
366
views
Convergence rate of Pearson correlation matrix
I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
- 121
2
votes
1
answer
663
views
Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets
Let $\Omega$ be an uncountable set and $(\Omega, \mathcal{F},P)$ be a probability space built on $\Omega$.
Let $S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty$ be a finite subset of ...
user42192
2
votes
0
answers
199
views
CLT for a Markov Renewal Process
Suppose $(X,T)=\{(X_n,T_n)\}_{n\geq0}$ is a Markov renewal process, where $X$ is a finite-state, discrete-time Markov chain with state space $\{1,2,...,R\}$. $T$ is the additive component, more ...
- 41
1
vote
2
answers
498
views
Show convergence result
Consider the following sets:
$$
A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\},
$$
and
$$
A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)...
- 108
1
vote
1
answer
193
views
Compute limit of $\mathbb P(Y \le X_n)$ using limiting information on the sequence of random variables $X_n$
Let $Y$ be a symmetric random variable, $(X_n)_n$ be a sequence of nonnegative random variables, and set $p_n = \mathbb P(Y \le X_n)$. It is known from Slutsky's theorem that, if $c$ is a constant ...
- 6,853
1
vote
1
answer
363
views
limit and combinatorics
Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit:
$\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$
When $...
1
vote
2
answers
355
views
Reference request and clarification for Central Limit Theorem for complex random variables
I'm looking for a reference and a proof of the following version (or eventually a more general version) of the Central Limit Theorem for complex random variables.
Theorem. Let $Z_1, Z_2, \dots, Z_n$ ...
- 361
1
vote
2
answers
169
views
Asymptotic properties of weighted random walks / infinite convolutions of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of
$$
\sum_{k=1}^n c^k X_k.
$$
I can prove that this ...
1
vote
1
answer
108
views
Maximum of the periodogram of a truncated sequence
Suppose that $Z,Z_1,Z_2,\ldots$ are iid random variables such that $\operatorname EZ=0$, $\operatorname EZ^2=1$ and $\operatorname E|Z|^s<\infty$ with some $s>2$. Let $\tilde Z_t=Z_tI_{\{|Z_t|\...
- 423
1
vote
1
answer
253
views
convergence for series of random variables
Suppose $X_n\sim N(0,1) $ is iid, then it is easy to see that
$$\sum_{n=1}^{\infty}\frac{X_n}{n}\cos nx$$
converges a.s. for any $x$ since
$$\sum_{n=1}^{\infty}var(\frac{X_n}{n}\cos nx)<\infty$$
...
- 125
1
vote
1
answer
156
views
How to show that $\int x \,d\nu = 0$ using a pseudo-weak convergence of measures?
I have a sequence of $p$-dimensional infinitely divisible random vectors $S_n'$, such that $S_n' \Longrightarrow X$, as $n \to \infty$.
Suppose the following assumptions
The characteristic functions ...
- 13
1
vote
1
answer
195
views
CDF of sum of independent cosines?
Consider the random variable
$$X=\frac{1}{d}\sum_{k=1}^d\cos X_k$$
where $X_k$ are each drawn uniformly i.i.d. from $[0,2\pi]$. What is the CDF of X?
It seems that a relatively direct way could be to ...
- 465
1
vote
1
answer
233
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
vote
1
answer
412
views
Almost sure convergence of the supremum over a class of random variables
Let $\mathcal{X}_n=\{ X_{n,\lambda}, \lambda \in \Lambda\}$ be a collection of random variables (defined on the same probability space) indexed by a deterministic index $\lambda$ over an index space $\...
- 195
1
vote
1
answer
475
views
Convergence of quadratic form $y^T Q y$ where $y$ is a random iid sequence of length $n$ and $Q$ is an $n \times n$ random matrix independent of $y$
For each positive integer, let $Q_n=(q_{i,j})_{i,j \in [n]}$ be a random $n \times n$ psd matrix. In the limit $n \to \infty$, suppose the eigenvalues of this sequence of matrices are uniformly ...
- 6,853
1
vote
1
answer
368
views
Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?
The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions :
Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...
- 128k
1
vote
1
answer
165
views
If $\mu_t\to\mu$ weakly, then $\limsup_t|\mu_t|(A)\le|\mu|(A)$ for all closed $A$
Let $E$ be a metric space, $\mathcal M(E)$ denote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$, $(\mu_t)_{t\in I}$ be a net ...
- 167
1
vote
2
answers
194
views
Continuity of the densities of a stochastic process
Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ an interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\mathbb{R}^d)$ for ...
- 463
1
vote
1
answer
480
views
Central limit theorem and convergence of means [closed]
If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that $\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\stackrel{d}{\to} \mathcal{N}(0,1),$ so ...
- 1,269
1
vote
1
answer
157
views
Is finding the CDF from the Laplace transform well-posed?
In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
- 654
1
vote
1
answer
114
views
Convergence in probability of sample covariance for permutation invariant triangular arrays
Take two triangular arrays $X_{N,i}$ and $Y_{N,i}$ of random variables where $1 \le i \le N$. Suppose that the families $\{X_{N,i}\}$ and $\{Y_{N,i}\}$ are independent, and that the following ...
- 1,124
1
vote
1
answer
107
views
Convergence of discretized process when its predictable part converges to infinite variation process
This question seems to be related to Theorem IX.7.28 in J. Jacod and A. Shiryaev's Limit theorems for stochastic processes (2013), and it is very important to prove asymptotic properties of my ...
- 284
1
vote
1
answer
197
views
Rate of variance's decrease for the mean's distribution of infinite variance i.i.d. random variables
Consider a set of i.i.d. (positive) random variables $\{X_i\}_{i=1}^N$. Each variable $X_i$ has a distribution with finite mean but infinite variance. In particular, if $P_{X_i}(x)$ is the P.D.F. of ...
- 305
1
vote
1
answer
571
views
Approximate expectation of a random variable that is the logarithm of a function of a binomial
I want the expectation of the following random variable: $\log\left(\frac{X}{k-X}+\alpha \right)$ with $X \sim Bin_{(k-1),p}$ and $\alpha > 0$, Therefore I derived the Taylor Series:
\begin{...
- 89
1
vote
2
answers
287
views
Showing $o(1)$ convergence for ratio of successive binomial tail probabilities
For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that
$$
\frac{P(X>c)}{P(X>c-1)}=1-o(1)
$$
uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...
- 59
1
vote
1
answer
81
views
Convergence rate of $\operatorname E|\langle X,f_n\rangle|^p$
Suppose that $X$ is a random element with values in a separable Hilbert space $\mathbb H$ such that $\operatorname EX=0$ and $\operatorname E\|X\|^2<\infty$. Suppose that $f_1,f_2,\ldots$ form an ...
- 423
1
vote
1
answer
153
views
Rate of convergence of exponential of sample mean to exponential of first moment?
If $X_n \sim N(\mu,\sigma)$ and $T_n = \frac{1}{n}\sum_1^n X_i$
What is the rate of convergence of $e^{T_n}$ to $e^{\mu}$
user94231
1
vote
1
answer
231
views
Asymptotic behaviour of a mean
Fix $x>0$ and $c\in\mathbb{N}$. Let $f(x):=\frac{c}{4c-2+2x^2}$ and
$$m_N(x):=\frac{1}{N} \sum_{i=0}^{f(x)N} \log(\frac{c N}{2}-i(2c-1))$$
I'm pretty sure $m_N(x)\to\infty$ as $N\to\infty$.
I ...
- 293
1
vote
0
answers
55
views
Limit process of a sequence of Gaussian variables on mesh grid going to zero
Consider the interval $[0,1]$ and a partition $\mathscr{P}_n = \{ [t_i,t_{i+1}), \, i=1,\ldots,N_n \, : \, 0=t_0 < \ldots < t_{N_n} = 1\}$. Suppose that for all $i$ and $t \in [t_i,t_{i+1})$, we ...
- 141
1
vote
0
answers
36
views
Uniform distribution as argument for copula likelihood
I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...
- 33
1
vote
0
answers
170
views
Asymptotic distribution of L infinity norm of Gaussian random vector
Let $\mathbf{X}_n = (X_{n,1}, \ldots, X_{n,n})$ be a $n$-dimensional random vector with $N_n( \mathbf{0}_n, \boldsymbol{\Sigma}_n )$ distribution. The asymptotic distribution of the $L_\infty$-norm of ...
- 119
1
vote
0
answers
82
views
Central limit theorem with limit of functions
Suppose that
$$\sqrt{n}(X_n - \theta)\xrightarrow{d} X,$$
according to the delta method, we have
$$\sqrt{n}(g(X_n)-g(X))\xrightarrow{d} g'(\theta)X$$
when $g$ is differentiable.
My question is, if
$$\...
- 11
1
vote
0
answers
57
views
Convergence of stochastic linear recurrences
Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$).
Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously ...
- 183
1
vote
0
answers
96
views
Limit of alternating sum of factorial moments which diverge
Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that
$$
P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!}
$$
...
- 640
1
vote
0
answers
169
views
Normal numbers and law of the iterated logarithm
If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...
- 3,098
1
vote
0
answers
430
views
Convergence in law and distribution theory
A standard result in probability theory asserts that a sequence of probablity measures $\mu_n$ on the Borel $\sigma$-algebra of $\bf R$ converges in law or weakly to a probability measure $\mu$ if and ...
- 18.7k
1
vote
0
answers
197
views
Weak convergence of Cesaro means of weakly converging infinite-dimensional distribution
Suppose we have sequences of random variables $\{X_{n,m},n \in \mathbb{N}\}$ where the distribution of $(X_{n,m})_{n\in\mathbb{N}}$ converges weakly to an infinite-dimensional normal distribution $\...
- 19
1
vote
0
answers
103
views
Convergence result on Cornish Fisher expansion of binomial distribution
Since it is known that Cornish Fisher expansion of quantiles does not have guaranteed convergence for all distribution, I wonder specifically if any convergence result is known in literature for CF ...
- 53
1
vote
0
answers
62
views
Reference request for invariance principles
In various places, an example being
https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254,
the authors consider a discrete-time process (real-valued, say) $(X_n)_{n \in \mathbb{N}}$, define ...
- 111
1
vote
0
answers
61
views
Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
- 31
1
vote
0
answers
131
views
Almost sure stochastic equicontinuity
Suppose $\mathcal{G}$ is a normed closed class of functions with finite entropy and envelope with a finite second moment (details below), and $g_0$ is a function in the interior of that class. Let $...
- 59
1
vote
1
answer
335
views
Finding a connection between two types of convergence
Please, help me find connections between two types of convergence:
Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences:
1) $X_n \...
- 31
1
vote
0
answers
87
views
Conditonal convergence implies convergence?
Note : All measures below are probability measures.
Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$.
Actually,...
- 245