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4 votes
1 answer
1k views

Quantile convergence

Let $X^1,\dots,X^n$ be a sample of (not necessarily iid) random variables and denote $$F^n(x)=\frac{1}{n}\sum_{i=1}^n \mathbf 1_{X^i\leq x}$$ the empirical distribution function. Suppose that we know ...
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 ...
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 ...
-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 ...
16 votes
0 answers
310 views

Randomized Pascal's triangle: What is the average of all the numbers?

This question was posted on MSE. It received some interesting responses, but no definite answer. Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for ...
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 \...
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-...
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 $...
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 ...
-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 ...
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 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 ...
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 ...
6 votes
1 answer
809 views

Probability that a positive integer is in the range of the Euler phi function

Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$. Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$. Is $\limsup_{n\...
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 $\...
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 $\...
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)...
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 ...
7 votes
1 answer
556 views

A variation on the Borel–Cantelli lemma theme

Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let \begin{equation*} E:=\bigcap_{n\ge0}B_n, \end{equation*} where \begin{equation*} B_n:=\...
27 votes
5 answers
3k views

How to show a function converges to 1

Consider the following recurrence relation in two variables: $$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$ for positive integers $a$ and $b$, with the boundary conditions $f(0,b)=0$ ...
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, ...
3 votes
1 answer
561 views

On the convergence in total variation

$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R$...
3 votes
2 answers
387 views

Definition of weak conditional convergence of random variables

I am looking for a definition of conditional convergence. Suppose that $X_1, X_2, \dots, X_n$ are $\mathbb R$-valued random variables with finite second moments, and $W_1, W_2, \dots, W_n$ are iid $\...
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-...
5 votes
0 answers
184 views

Question about $n$ random points in a regular polygon, and a limiting probability

Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...
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]{\...
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 ...
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 $\varepsilon >0$ ...
2 votes
2 answers
298 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 ...
4 votes
1 answer
205 views

Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$

Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...
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 $$\...
1 vote
2 answers
356 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 $Z_1, Z_2, \dots, Z_n$ ...
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 ...
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 $y_0, u_1, u_2,...$ ...
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 $$\sum_{n=1}^{\infty}var(\frac{X_n}{n}\cos nx)<\infty$$ ...
6 votes
2 answers
775 views

Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped

I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
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 ...
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 ...
4 votes
1 answer
2k views

Examples of convergence in distribution not implying convergence in moments

It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions. Let $\{X_n\}$ be a ...
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 ...
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 ...
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 $...
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
234 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....
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 $\...
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 ...
3 votes
1 answer
397 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 ...
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 ...
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 ...
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 ...