All Questions
Tagged with limits-and-convergence pr.probability
184 questions
3
votes
1
answer
190
views
Quantitative version of ergodic theorem in Markov chains
Consider an irreducible Markov chain $X_t$ with finite state space $E$, and unique invariant measure $\pi$. Fix a function $V:E\to\mathbb R$ such that $E_\pi[V]=0$. The ergodic theorem tells us that, ...
- 59
3
votes
1
answer
829
views
The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$
In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
user157884
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
125
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 ...
- 5,405
3
votes
1
answer
182
views
Superlinear Convergence of a Markov Chain
Suppose that we have a Markov process $\{Z_t\}_{t=0}^\infty$, where $Z_t \geq 0$ for any $t$. Assume that, conditioning on $Z_t = z_t$, we have
$
\mathbb{E}\{Z_{t+1}|Z_t = z_t\} \leq \kappa z_t^2
$. ...
- 1,127
3
votes
1
answer
1k
views
Book on Convergence Concepts in Probability without Measure Theory [closed]
I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
- 319
3
votes
2
answers
505
views
Precise asymptotics for moments of order statistics of normal distribution
Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...
- 151
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
3
votes
1
answer
196
views
Uniform Convergence for Vectors
$\textbf{Problem statement:}$
Let $\mathcal H:\mathcal X \rightarrow \{0,1\}$ be a class of Boolean functions for $\mathcal X \subset \mathbb R^n$, and let the VC Dimension of $\mathcal H$ be $VC_{...
- 197
3
votes
1
answer
751
views
Wasserstein convergence of conditional measures
Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the same atoms):
...
- 710
3
votes
2
answers
227
views
Liminf of the maximum of two iid sequences
Let $\{X_t\}_{t\ge1}$ and $\{Y_t\}_{t\ge1}$ be two iid sequences of random variables that have full support. That is, if $A\subseteq\mathbb{R}$ has positive Lebesgue measure, then $P(X\in A) >0$ ...
- 479
3
votes
1
answer
400
views
Countable convergence-determining class for weak convergence of probability measures
Suppose that $E$ is a Polish space.
Portmanteau theorem asserts that a sequence $(\mu_n)$ of Borel probability measures weakly converges to a Borel probability measure $\mu$ (shortly, $\mu_n\overset{...
- 33
3
votes
1
answer
177
views
Convergence of SDEs
Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
- 5,405
3
votes
1
answer
377
views
Law of large numbers for random functions?
Is there a version of the law of large numbers for random functions of the type: $h(X_j,\hat{\theta}_n)$, where $X_1,\dots,X_n$ are i.i.d. random variables, with distribution $F$, and $\hat{\theta}_n =...
- 31
3
votes
1
answer
354
views
Central Limit Theorem for additive function of permutations of sequences
Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational.
For each $n$ such ...
- 325
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
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
105
views
How to show monotonocity and the limit? [closed]
Let me reformulate my recent question.
Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density:
$$\phi(x) = C\left\{ \begin{array}{lcc}
\sqrt{...
- 31
3
votes
0
answers
157
views
Pointwise convergence of ergodic averages of unconventional conditional expectations
Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ($n,...
- 191
3
votes
0
answers
567
views
Berry-Esseen result for triangular arrays
Let $\{\{X_{n1},\ldots,X_{nn}\},n=1,2,\ldots\}$ be a triangular array of independent random variables where each row contains identically distributed random variables. Let $E[X_{n1}]=\mu_n<\infty$ ...
- 917
3
votes
0
answers
85
views
integral against a gaussian density over an increasing space
Consider the following Gaussian density over $\mathbb{R}^{2^n}$
$$p_n(\underline{x}):=\frac{\exp(-\frac{1}{2n}\langle C_n^{-1}\underline{x},\underline{x}\rangle)}{\sqrt{(2\pi n)^{2^n} \det C_n}}$$
...
- 293
2
votes
1
answer
224
views
Approximate size of the image of functions $f:[n]\to[n]$ [closed]
The following is inspired from the most recent riddle of the week of the German news magazine Der Spiegel.
For any positive integer $n\in\mathbb{N}$, let $[n]$ denote the set $\{1,\ldots,n\}$. Let $...
- 48.9k
2
votes
1
answer
351
views
Lower bound on sum of independent heavy-tailed random variables
I have a sum of $n$ i.i.d random variables $X_i$ such that $E[X_i] = 0$,$\mathrm{E}[|X_i|^{1 + \delta}]$ exists for some $0 < \delta < 1$ but $\mathrm{E}[|X_i|^{1 + \delta+ \epsilon}]$ does not ...
- 23
2
votes
1
answer
131
views
Almost sure convergence of double averages of IID random variables
Let $ \{X_i\}_{i=1}^{P} $ and $ \{Y_j\}_{j=1}^{Q} $ be two sequences of independent and identically distributed (i.i.d.) random variables. $X_i$ and $Y_j$ are independent between all pairs of $i$ and $...
- 433
2
votes
1
answer
196
views
Pointwise almost sure convergence implies global convergence
Sorry in advance if this is not sufficiently research-level, it is really more of a reference request since the proof is not difficult. Let $\mathcal{Y}$ be a compact set, let $\{X_n\}$ denote a ...
- 4,049
2
votes
2
answers
1k
views
Convergence in probability of series of random variables
From the standard literature it is well known that for sequences of random variables $X_{1, n} \stackrel{P}{\rightarrow} X_1$ and $X_{2, n} \stackrel{P}{\rightarrow} X_2$ as $n \rightarrow \infty$ it ...
- 299
2
votes
1
answer
268
views
Convergence in probability of a supremum
Let $A>0$ be fixed and consider $X_1,\ldots$ i.i.d. nonnegative random variables such that $E[1/X_1]<\infty$.
Is is true that $$\sup_{a\in \big (0,\frac A{\sqrt n} \big]} \sum_{i=1}^n 1_{X_i>...
- 123
2
votes
1
answer
383
views
Lower bound and limit of a sum with binomial coefficients
Let
$$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$
$$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$
$$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...
- 155
2
votes
1
answer
185
views
Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$
Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.
...
- 6,853
2
votes
2
answers
297
views
Convergence of the row sums in a triangular null array with zero mean
Let $(X_{jn})_{1\leq j \leq n}$, $n\in \mathbb N$, be a triangular array of random vectors in $\mathbb R^d$ (the $X_{jn}$ are understood to be independent in $j$ for fixed $n$.). We say that the ...
- 13
2
votes
1
answer
120
views
Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes
Let $X = [X_t : t \in \mathbb{Z}] \sim P$ and $Y = [Y_t : t \in \mathbb{Z}]\sim Q$ be two stochastic processes. Let's define the Mallows metric. Let $\mathcal{M}_m$ be the random vectors $(X,Y)$ ...
- 135
2
votes
1
answer
403
views
Law of large numbers for triangular arrays whose moments "look independent"
Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that
$$
\mathbb{E}\left[\...
- 1,124
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
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
99
views
Convergence of estimator given by a fixed point
Let $X$ be a non-negative random variable with cdf $F$ and define
$$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function.
Let $s_0$ be the unique fixed point of $G$.
Now let $X_1,\dots,X_t$ ...
- 23
2
votes
1
answer
254
views
Asymptotic rate for the expected value of the square root of sample average
I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$.
Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$.
I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$.
My first ...
- 167
2
votes
2
answers
152
views
Divergence rate of geometric sum of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
$$
0<\lim_{\beta\rightarrow 1}(1-\...
- 479
2
votes
1
answer
643
views
Weak convergence in Skorohod space
I am reading a paper where they prove Donsker's invariance principle for a sequence of dependent RV's. They do the following steps which I can't follow so well. I won't write out the precise ...
- 21
2
votes
1
answer
74
views
Conditions for absorption
Let $X$ be a Markov chain with countable state space $S$ and transition kernel $P$, and let $h \colon S \to [0,1]$ be a sub-harmonic or super-harmonic function. Assume that for all $\varepsilon >0$ ...
user514580
2
votes
1
answer
184
views
A question about convergence of stochastic processes converging to a random walk
Consider the following random walk $(y_t)_{t \in \mathbb Z_+}$:
$$y_t = y_{t-1} + u_t,\quad (u_t)_{t \in \mathbb Z_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z_+)$$
where $y_0, u_1, u_2,...$ ...
- 13
2
votes
1
answer
116
views
A question on the applicability Chebyshev inequality for sequence of random quantities
Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function.
...
- 6,853
2
votes
1
answer
124
views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
- 399
2
votes
2
answers
736
views
Submartingales bounded in $L^p$, $p>1$
Let $p>1$ be a real number. It is known that if $(X_n)_{n\geq 0}$ is a martingale bounded in $L^p$ (i.e. $\sup\{\mathbb{E}(|X_n|^p), n\geq 0\} < +\infty$ ), then $(X_n)_{n\geq 0}$ converges a....
user102214
2
votes
1
answer
444
views
Literature question on the convergence rate of the empirical distribution
Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each $...
user83947
2
votes
1
answer
305
views
Convergence of weighted double sum of random variables
I'm looking for convergence results of particular weighted sum:
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$
when random variables $X_i$ ar i.i.d. Are there any investigation ...
- 267
2
votes
1
answer
137
views
Variant of Skorokhod's theorem
Consider the following situation:
$S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful).
There is a a random variable $\zeta: \Omega \to S$.
$f_n(\zeta) \to^d \eta$, i....
- 2,221
2
votes
1
answer
315
views
Linear or quadratic combinations of i.i.d. random variables [closed]
I already posted this question here https://math.stackexchange.com/questions/769920/law-of-large-numbers-for-linear-quadratic-combinations-of-i-i-d-random-variab but I received no answers.
Let $(X_i)...
- 223
2
votes
1
answer
2k
views
Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?
Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...
- 23
2
votes
2
answers
328
views
Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
- 1,043
2
votes
0
answers
150
views
A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
- 657