All Questions
Tagged with pr.probability limits-and-convergence
20 questions
5
votes
3
answers
5k
views
Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere
Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
- 6,853
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
2
votes
2
answers
322
views
If $(\exp(\mu_n))_{n\in\mathbb N}$ is weakly convergent, is the normalized sequence convergent as well?
Let $E$ be a metric space and $\mathcal M(E)$ denoote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$.
I would like to know ...
- 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
0
votes
1
answer
159
views
Approximation of a random sum of random variables (infinitely divisible distribution) by a triangular array
We know that a Poisson distribution can be approximated by a binomial distribution. More exactly, let $(X_{jn})_{1\leq j \leq n}$ be a i.i.d. triangular array such that
$$P[X_{jn}= 1 ] = p_n = 1- P[X_{...
- 135
27
votes
4
answers
3k
views
Rate of convergence of $\frac{1}{\sqrt{n\ln n}}(\sum_{k=1}^n 1/\sqrt{X_k}-2n)$, $X_i$ i.i.d. uniform on $[0,1]$?
Let $(X_n)$ be a sequence of i.i.d. random variables uniformly distributed in $[0,1]$; and, for $n\geq 1$, set
$$
S_n = \sum_{k=1}^n \frac{1}{\sqrt{X_k}}\,.
$$
It follows from the generalized central ...
- 1,372
7
votes
1
answer
261
views
Comparison of several topologies for probability measures
Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
- 243
5
votes
4
answers
917
views
Limit of a sum with binomial coefficients
Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$
$$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$
$$C_k = \frac{\sum_{i=1}^k(...
- 155
5
votes
1
answer
4k
views
When is the limit of Martingales a Martingale?
I have a sequence of continuous time random variables $X_n(t)$ where $t \in [0,1]$. Suppose that there is a filtration $F_t$ such that for each $n$, $X_n$ is a martingale with respect to this ...
- 195
4
votes
2
answers
349
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 ...
3
votes
1
answer
473
views
Expected value of the maximum of the periodogram
Let us suppose that $X_1,\ldots,X_n$ with $n\ge1$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$ and define the DFT of $X_1,\ldots,...
- 423
3
votes
1
answer
607
views
Show that $\sup_{\|g\|\leq \delta_n}\left| \frac{1}{\sqrt{n}}\sum_{i=1}^n g(Z_i)\right|\rightarrow_{\text{a.s.}}0.$ when $\delta_n\rightarrow 0$?
UPDATE: The result below can be understood as an almost sure stochastic equicontinuity condition. I don't know of any result establishing primitives of almost sure stochastic equicontinuity. If you ...
- 59
3
votes
1
answer
162
views
Weak convergence of Dirichlet distributions to a "multi-Bernoulli" distribution
For a positive vector $\alpha\in\mathbb{R}^n$ ($n\geq 1$), denote by $\text{Dir}(\alpha)$ the Dirichlet distribution with parameter $\alpha$. In terms of weak convergence, is it true that, if $\sum\...
- 449
2
votes
2
answers
736
views
Submartingales bounded in $L^p$, $p>1$
Let $p>1$ be a real number. It is known that if $(X_n)_{n\geq 0}$ is a martingale bounded in $L^p$ (i.e. $\sup\{\mathbb{E}(|X_n|^p), n\geq 0\} < +\infty$ ), then $(X_n)_{n\geq 0}$ converges a....
user102214
2
votes
1
answer
101
views
If signed measures $\mu_n$ are such that $\mu_n\to\mu$ and $\|\mu_n\|\to c\in(0,\infty)$, does $\exp^*(\mu_n)/\|\exp^*(\mu_n)\|$ necessarily converge?
$\newcommand{\R}{\mathbb R}$Let $M$ denote the set of all finite signed measures on a separable Banach space $B$. For any $\mu\in M$, let
\begin{equation*}
\exp^*(\mu):=\sum_{k=0}^\infty\frac{\mu^{...
- 128k
2
votes
1
answer
124
views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
- 399
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
0
votes
1
answer
169
views
Understanding the approximation of a random sum of random processes
I want to understand an approximation of a compound Poisson distribution in this paper.
First, let's set the environment. Consider $\mathcal{P}$ the class of distributions of real-valued and strictly ...
- 135
0
votes
1
answer
450
views
A complex question related to a certain convergence of Lévy measures
Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
- 13
0
votes
1
answer
197
views
Bound the expectation of an average
Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...
- 108