All Questions
Tagged with pr.probability limits-and-convergence
184 questions
2
votes
1
answer
120
views
Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes
Let $X = [X_t : t \in \mathbb{Z}] \sim P$ and $Y = [Y_t : t \in \mathbb{Z}]\sim Q$ be two stochastic processes. Let's define the Mallows metric. Let $\mathcal{M}_m$ be the random vectors $(X,Y)$ ...
- 135
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
2
votes
1
answer
403
views
Law of large numbers for triangular arrays whose moments "look independent"
Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that
$$
\mathbb{E}\left[\...
- 1,124
4
votes
1
answer
87
views
Distance between trunctated random walk and its normal form
I have $$X_i \sim N(0,1), \quad S_n=X_1+\cdots+X_n,$$
$$ \mathscr{S}_n (t, \omega) := \frac{1}{ \sqrt{n} } \sum_{i=1}^{n} \left[ S_{i-1} (\omega ) + n \left( t - \frac{i-1}{n} \right) X_{i}(\omega) \...
- 43
3
votes
1
answer
271
views
For a random sequence $X_0, X_1, X_2, \ldots$ and $F_n$ the empirical CDF, does $F_n(X_0)$ converge to a uniform random variable?
Let $X_0, X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous CDF $F(x)$ and define a sequence of ...
- 277
2
votes
1
answer
383
views
Lower bound and limit of a sum with binomial coefficients
Let
$$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$
$$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$
$$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...
- 155
5
votes
4
answers
917
views
Limit of a sum with binomial coefficients
Let $$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
4
votes
2
answers
783
views
Convergence almost everywhere of characteristic functions
Let $(\Phi_n)_n$ be the characteristic functions of probability measures $(\mu_n)_n$ and let $\Phi$ be the characteristic function of a probability measure $\mu$.
Do you know an example where $\Phi_n(...
- 93
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
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
2
votes
1
answer
196
views
Pointwise almost sure convergence implies global convergence
Sorry in advance if this is not sufficiently research-level, it is really more of a reference request since the proof is not difficult. Let $\mathcal{Y}$ be a compact set, let $\{X_n\}$ denote a ...
- 4,049
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
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
2
votes
1
answer
116
views
A question on the applicability Chebyshev inequality for sequence of random quantities
Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function.
...
- 6,853
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
2
votes
2
answers
1k
views
Convergence in probability of series of random variables
From the standard literature it is well known that for sequences of random variables $X_{1, n} \stackrel{P}{\rightarrow} X_1$ and $X_{2, n} \stackrel{P}{\rightarrow} X_2$ as $n \rightarrow \infty$ it ...
- 299
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
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
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
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
7
votes
1
answer
259
views
Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
- 128k
2
votes
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
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
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
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
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
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
4
votes
3
answers
914
views
Sample average L1 convergence speed
Say $X_1, \cdots, X_n$ are i.i.d random variables with mean zero, let $S_n = \sum_{i=1}^n X_i$, we know by SLLN $$\frac{S_n}{n}\rightarrow 0\text{ a.s}$$
We could further know that the sequence of ...
- 173
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
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
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
-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 $$\left.\lambda\right|_\delta(B):=\...
- 167
0
votes
0
answers
302
views
Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
1
vote
1
answer
571
views
Approximate expectation of a random variable that is the logarithm of a function of a binomial
I want the expectation of the following random variable: $\log\left(\frac{X}{k-X}+\alpha \right)$ with $X \sim Bin_{(k-1),p}$ and $\alpha > 0$, Therefore I derived the Taylor Series:
\begin{...
- 89
0
votes
0
answers
74
views
Convergence of stochastic process $X_n$
Consider the discrete time random process $X_n,n\in \mathbb N$, with
$$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
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
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
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
156
views
When does a gaussian quadratic form converge (in probability) to a constant?
Let $(h_{ij})_{i,j \in \mathbb N}$ be a sequence of real numbers (deterministic) and let $x_1,\ldots,x_n,\ldots$ be a sequence of iid $N(0,1)$ randm variables. For each positive integer $n$, consider ...
- 6,853
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
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
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
3
votes
0
answers
136
views
An integral involving Levy process with no positive jumps
Let $L_t$ be a Levy process that has no positive jumps, but is not strictly decreasing, i.e
$$
L_t = \gamma t + \sigma B_t + J_t,
$$
where $B_t$ is a Brownian motion, $J_t$ is a pure jump process with ...
- 103
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
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
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
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
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