All Questions
Tagged with pr.probability limits-and-convergence
184 questions
1
vote
0
answers
88
views
Questions about generalized Polynomial Chaos, book by Dongbin Xiu
I have some questions about Chapter 5 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu.
Theorem 5.7: Let $Y$ be a random variable and $\mathbb{E}[Y^2]<\infty$. Let $Z$ ...
- 31
1
vote
0
answers
202
views
Convergence in probability in the setting of free probability
Let $A_n$ and $B_n$ be sequences of positive-definite random matrices whose empirical spectral distributions converge to (possibly different) limiting spectral distributions $\mathcal A$ and $\mathcal ...
- 321
1
vote
0
answers
218
views
Exponential Ergodicity for Reflected Brownian Motion in a Bounded Domain
Assume we have a reflected Brownian motion in a smooth bounded domain $D \subseteq \mathbb R^d$. It can have nonzero (but constant) drift, non-identity (but constant) covariance matrix, and oblique (=...
- 45
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
0
votes
1
answer
201
views
Infinite limit of sums of gamma functions is constant?
The following expression arises in the study of hierarchical models. I suspect that the sum of the underlined $4$ terms become constant as $\alpha\rightarrow \infty$. Mathematica agrees when prompted ...
- 111
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
0
votes
2
answers
182
views
Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)
Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
\mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...
- 135
0
votes
1
answer
376
views
Random variable is Big O in probability notation
Let $R_n$ be a random variable with values in $[0,1]$ and $nR_n$ converges to $\frac{1}{1+C} \chi_m^2$ in distribution for some constant $C>0$ and $m\in \mathbb{N}$. Is it possible to show that $(1-...
- 115
0
votes
2
answers
204
views
What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?
Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges ...
- 6,853
0
votes
1
answer
127
views
Weak convergence to a "multi-Bernoulli" distribution
Let $(X_n)_{n\geq 1}$ be a sequence of random variables defined on the $d-$simplex ($d\geq 1$) : $\Sigma_d=\big\lbrace\boldsymbol{x}\in\mathbb{R}_+^d,\,\sum_{1\leq i\leq n} x_i=1\big\rbrace$. Assuming ...
- 449
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
0
votes
1
answer
72
views
Exceedance distribution of Levy process
Consider a Levy process $L(t)$ with linear drift $-1$, no Brownian motion component, and Poisson jumps at rate 2 with size Uniform($0, 1$), and with $L(0)=0$. This process has zero mean drift.
Let $\...
- 294
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
478
views
Covariance in the limit of random variables
Suppose $\{X_n\}$ and $\{Y_n\}$ are two sequences of random variables and we know that $X_n \overset{L^2}{\to} X$ and $Y_n \overset{L^2}{\to} Y$, where $\overset{L^2}{\to}$ means converge in mean ...
- 123
0
votes
1
answer
4k
views
Convergence of the empirical distribution function
Let $\alpha\in\mathbb R^d$, with $\alpha\neq 0$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $R$ the cdf of $\alpha X^1$ (where the product is a ...
- 293
0
votes
1
answer
145
views
Limiting probability question [closed]
Let $X$ denote an $m\times n$ matrix and suppose that each value $x_{ij}$ is an integer that is selected uniformly at random from ${1,\dots,n}$, independently of all other values. If we fix $m$ and ...
0
votes
0
answers
134
views
Asymptotics of a ratio on the unit sphere
Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$.
Consider the ratio (for $k \geq n$)
$$
R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-...
- 420
0
votes
0
answers
41
views
Formalization of sample convergence
Let's say I have a sample of $X_1, \dots, X_n$, where I know that $X_i$ were generated by some ARCH(1) process. It means that
$$X_i = \sigma_i z_i,$$
where $z_i \stackrel{iid}{\sim} N(0, 1)$ and $\...
- 33
0
votes
0
answers
76
views
Convergence of probabilities imply convergence of joint probability
Context: Suppose I have two pairs of sequences of random variables $X_n, \tilde{X}_n$ and $Y_n, \tilde{Y}_n$, where $X_n$ and $Y_n$ are not necessarily independent for any $n$, but $\tilde{X}_n$ and $\...
- 33
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
0
answers
67
views
LLN of random nearest neighbor function
There are two samples of iid random variates: $X=\{X_1,X_2,...,X_n\}$ and $Y=\{Y_1,Y_2,...,Y_n\}$. Further, $\forall i,j: X_i$ is independent of $Y_j$. The probability distributions $P,Q$ are unknown ...
- 89
0
votes
0
answers
202
views
$|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ under $E(|X_1|)<\infty$?
For i.i.d. random variables $X_1,\dots, X_n$ with $E(|X_1|)<\infty$. Does the following equation hold?
$$
\left|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)\right|=O_P\left(\frac{1}{\sqrt{n}}\right)
$$
I ...
- 193
0
votes
0
answers
302
views
Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
0
votes
0
answers
74
views
Convergence of stochastic process $X_n$
Consider the discrete time random process $X_n,n\in \mathbb N$, with
$$X_{n+1}=(1-K)\cdot X_n+K\cdot\frac{G_n}{c}\cdot X_n$$
where $G_n$ is a random variable with expectation $\mathbb E[G_n\mid X_n]=\...
- 5
0
votes
0
answers
156
views
Total variation convergence of random matrices and convergence of empirical spectral distributions
In the paper https://arxiv.org/pdf/1411.5713.pdf, on page 17, the authors prove in Theorem 7 that the total variation distance between the joint distribution of the entries of certain Wishart matrices ...
- 1
0
votes
0
answers
146
views
Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
0
votes
1
answer
220
views
Distributions associated with random sets and sums of random sets
Let's say you have an infinite random set $S$ of non-negative integers, and $T=S+S=\{x+y$ with $x,y\in S\}$. Let $N_S(z)$ be the number of elements of $S$ less than or equal to $z$; it is a random ...
- 3,098
0
votes
0
answers
123
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
0
votes
0
answers
691
views
$L_1$ convergence for a product of indicator functions
Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions
$$
\lim_{N\...
- 479
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 125
-1
votes
1
answer
80
views
Seating assignment inspired question
Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course ...
- 48.9k
-1
votes
1
answer
103
views
Convergence in $\mathbb{L}_1$ implies convergence "perturbed" conditional expectations
Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following ...
- 141
-1
votes
1
answer
396
views
Convergence of Radon-Nikodým derivative
Imagine we have a sequence of finite measures $\nu_n << \mu_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have ...
- 183
-2
votes
1
answer
108
views
If a sequence of measures is weakly convergent outside each compact ball, the sequence itself is weakly convergent
Let $E$ be a $\mathbb R$-Banach space and $\mathcal M_+(E)$ denote the space of finite nonnegative measures on $\mathcal B(E)$.
If $\lambda\in\mathcal M_+(E)$, let $$\left.\lambda\right|_\delta(B):=\...
- 167