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Questions tagged [probability-limit-theorems]

For question about limit theorems of probability theory, like the law of large numbers, central limit theorem or the law of iterated logarithm.

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Expectation of max of geometric r.v.

Let $(X_i)_{i \geq 1}$ be a sequence of i.i.d. geometric random variables such that $X_i \sim \text{Geom}(1/2)$ and $\mathbb{E}[X_i]=1$. I want to show that $$\lim_{n \rightarrow \infty} \frac{1}{n^2}\...
Mathick's user avatar
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1 vote
0 answers
23 views

Proof of Strong Consistency of mean of Beta posterior distribution

Suppose that we have random variable $X_{1}, X_{2}, ..., X_{n} \sim^{iid} \text{Bernoulli}(p_{0})$ with $p_{0}$ true unknown probability in $[0,1]$. Now, I want to implement Bayesian machinery to ...
Jonathan1234's user avatar
  • 1,083
1 vote
0 answers
38 views

exchange order of limit of random variable

Suppose to have a sequence of discrete random variables $X_n(\lambda)$ depending on some parameter $\lambda \ \in [0,\infty]$ and $n \in \mathbb{N}$ and that the following limits hold almost surely: $...
Riccardo's user avatar
1 vote
0 answers
18 views

From quenched limit theorems to annealed ones

Consider a discrete time process $X = (X_n)_{n \geq 0}$ in a random environment. If I can show that for every realisation of the random environment (i.e. under the quenched measure) $X_{\lfloor{nt}\...
Mathick's user avatar
  • 318
0 votes
1 answer
44 views

Showing why Poisson random variable is not sub gaussian

I looked at a previous answer posted on stackexchange; Poisson random variable is not sub gaussian; however, I cannot see how to obtain a lower-bound using Stirling's approximation. For context, I am ...
Decaying Tails's user avatar
2 votes
1 answer
54 views

Does the limit of $\binom{n}{np} p^{np} (1-p)^{n(1-p)}$ exist when $n\rightarrow +\infty$ with a constant $p\in [0, 1]$?

Does the limit of $\binom{n}{np} p^{np} (1-p)^{n(1-p)}$ exist when $n\rightarrow +\infty$ with a constant $p\in [0, 1]$? Intuitively, we consider a box with a proportion $p$ of balls being red and the ...
Yuzhen Feng's user avatar
3 votes
0 answers
52 views

Convergence of quotient of sample variance and sample mean

Let $(X_k)$ be a collection of i.i.d random variables with finite fourth moment satisfying $\mathbb{E} X = 0$ and $\mathbb{E} X^2 = 1$. What can be said about the convergence of the following quotient ...
Kayle of the Creeks's user avatar
2 votes
2 answers
103 views

Proportion of vertices in components of size $k$ in Erdos Renyi

Consider $G(n,c/n)$ the Erdos-Renyi graph on $n$ vertices with the probability of having an edge between any two vertices is $c/n$. Let $X_{n,k}$ be the proportion of vertices in size-$k$ components. ...
Sergio's user avatar
  • 104
3 votes
1 answer
60 views

Finding $\lim_{\,n\to\infty} \, \sum_{k=n}^{3n} \binom{k-1}{n-1}\left(\frac{1}{3}\right)^n\left(\frac{2}{3}\right)^{k-n}$

The problem is to find the following limit: $$\lim_{\,n\to\infty} \, \sum_{k=n}^{3n} \binom{k-1}{n-1}\left(\frac{1}{3}\right)^n\left(\frac{2}{3}\right)^{k-n}$$ I know it has something to do with CLT ...
Aleksi's user avatar
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2 votes
0 answers
32 views

Limit of stochastically continuous process

Let $(X_n)$ be a sequence of random variable taking values in $\mathbb{D}[0,1]$, the space of cadlag function equipped with Skorokhod topology, so that each $X_n$ can be viewed as a cadlag stochastic ...
George's user avatar
  • 105
0 votes
0 answers
13 views

Normal approximation to Poisson weighted sum

I am trying to come up with a way to approximate the sum given below by a normal distribution. $$ P(F| \overline{n}) = \sum_{n=0}^{\infty}P(F|n)P(n|\overline{n}) $$ F is the sum of IID random ...
John Smith's user avatar
3 votes
1 answer
185 views

Uniform Tightness of a Sequence of Probability Measures in $\mathbb R^{\mathbb N}$

Consider a sequence $(P_n)_{n\in\mathbb N}$ of probability measures on $(\mathbb R^{\mathbb N}, \mathcal B)$, where $\mathbb R^{\mathbb N}$ is the countable Cartesian product of $\mathbb R$, and $\...
Syd Amerikaner's user avatar
1 vote
1 answer
43 views

If a random variable converges to zero in probability what can we say about its almost sure boundedness?

First let me start with definitions that I will be using in the question. A sequence of random variables $X_n(\omega)$ converges to zero in probability if for any $\epsilon>0$, and any $\delta>...
curiosity's user avatar
  • 151
2 votes
1 answer
62 views

Probability of the limit equals the limit of the probability?

First, let me define the limit of a sequence of events, because I don't know how standard it is. Let $\Omega$ be a set and $\{ A_n \}_{n\in\Bbb N}$ be a sequence of sets s.t. $A_n\subseteq\Omega \ \...
Pablusky's user avatar
0 votes
2 answers
50 views

show that if $(\sqrt{n}(Y_n - \theta) \overset{d}{\to} N(0, 1))$ then $(Y_n \overset{P}{\to} \theta)$

I have the following question: Show that if $(Y_n)$ be a sequence of random variables that satisfies $(\sqrt{n}(Y_n - \theta) \overset{d}{\to} N(0, 1))$ then $(Y_n \overset{P}{\to} \theta)$. I've ...
ZedVeZed's user avatar
1 vote
1 answer
48 views

An example application where convergence in probability is useful or valuable

I have been learning different notions of convergence for sequences of random variables. I know the almost sure convergence is the strongest. The next best thing is convergence in probability. Can ...
curiosity's user avatar
  • 151
1 vote
0 answers
50 views

How to prove that $X_n \stackrel{\text{a. s.}}{\rightarrow} 0$

Let $\left\{X_n\right\}_{n \in \mathbb{N}}$ be a strictly decreasing sequence of positive random variables such that $X_n \stackrel{P}{\rightarrow} 0$. I have to prove that $X_n \stackrel{\text{a. s.}}...
Cyclotomic Manolo's user avatar
0 votes
0 answers
22 views

Measuring Robustness in Variational Bayesian Inference and Nonlinear Filtering

I am interested in how to properly pose/measure robustness, in a qualitative or potentially quantitative manner, when inferring a probability density function (pdf) either by Bayes' rule or a ...
kjc93's user avatar
  • 41
0 votes
1 answer
53 views

Convergence of bootstrap variance estimator [closed]

Let $X = X_1, \dots, X_n$ be a random sample and let $X^* = X_1^*, \dots, X_n^*$ be a bootstrap sample drawn from $X$ with replacement. Say $\hat \sigma_n^2$ is a consistent variance estimator, i.e., $...
Lime91's user avatar
  • 151
2 votes
0 answers
24 views

Convergence in distribution of random variables with symmetric distribution

I found the following question in the probability course materials: let $ (X_n)_{n\geq 1} $ be i.i.d. random variables with symmetric distributions such that $\sigma^2=\mathbb{E}[X_1^2]<\infty$. ...
sgvozdic's user avatar
1 vote
0 answers
15 views

MLE and limit distribution of ratio of parameters

I am solving an estimation problem and I can't make any progress. I have $ (X_{i1},X_{i2})^T $ iid from $\mathcal{N}_2 ( \mu , \Sigma)$. Define $$ \lambda_j = \frac{\mu_j}{\sqrt{\sigma_{jj}^2}} , \...
daniel's user avatar
  • 753
1 vote
1 answer
17 views

Express the Chi and uniform distributions as the limit of some coin tossing

The exponential distribution is the continuous limit of the geometric. What would be the coin-toss equivalent of the Chi and the uniform distributions in the following sense? For a sequence $p_{1..\...
Carlos Pinzón's user avatar
4 votes
1 answer
438 views

Does convergence of measures, with respect to the KL-divergence, imply almost sure convergence of the RN-derivatives? [closed]

Suppose I have a space $(\mathbb{X}, \mathbb{B}_\mathbb{X})$ where $\mathbb{B}_\mathbb{X}$ is the Borel $\sigma$-algebra of $\mathbb{X}$. On said space, I have probability measures, $P,P_1,P_2,\ldots$,...
addition_man's user avatar
1 vote
1 answer
70 views

Almost surely convergence of Bernoulli distribution ($\frac{1}{n}$)

So I have the following exercise: Let $X_n$ independent Bernoulli with parameter $\frac{1}{n}$, so $$ P(X_n=1)=\dfrac{1}{n}, \quad P(X_n=0)=1-\dfrac{1}{n}.$$ Show that $X_n$ converges to $1$ almost ...
Elemer Kit's user avatar
6 votes
1 answer
85 views

Central limit theorem for two-sided Pareto distribution

I am trying to solve the following problem, which provides an example for a central limit theorem in spite of the fact that the variance is infinite. Consider the two-sided Pareto distribution with ...
EnergySkiller's user avatar
2 votes
1 answer
119 views

What is the limit probability an element of $x \in S$ belongs to $f^n(S)$, for $n \to \infty$?

Let $S$ be a finite set of $|S|=n$ elements and $F$ be the set of all functions $f:S\rightarrow S$. It's easy to demonstrate that the integer sequence $\{c_i\} = |{\rm Im}(f^i)|$: is non increasing; ...
Yuri S VB's user avatar
1 vote
0 answers
29 views

Convergence of empirical CDF $F_n(t-) \to F(t-)$ for $t \in \mathbb{R}$

I am aware of course of the law of large numbers asserting the convergence of the sample mean to its population counterpart (in probability or even almost surely). In the case of the empirical CDF $...
JohnK's user avatar
  • 6,494
1 vote
1 answer
36 views

Asymptotic distribution of MLE of $\sigma$ for $N(0,\sigma^2)$

I know that given $X_1,...,X_n \sim N(0,\sigma^2)$, the MLE for $\sigma$ is $\hat{\sigma} = \sqrt{\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^2}$. I want to find the asymptotic distribution for $\hat{\...
Saim Faigol's user avatar
1 vote
0 answers
36 views

Convergence of multivariate normal distributions

Consider a sequence of $k$-dimensional random vectors $\mu_n$ and $k \times k$ random matrices $\Sigma_n$ which are positive definite. For each realization of $(\mu_n,\Sigma_n)$, call it $(\mu^{(i)}_n,...
nervxxx's user avatar
  • 281
0 votes
0 answers
13 views

Interchanging infinite sum and limit in distribution

I'm trying to do a proof for a project and I've run into the following problem. For each $j$ consider a sequence $(Y_{j,n})_{n \in \mathbb{N}}$ of random variables such that the different sequences ...
Snildt's user avatar
  • 376
1 vote
1 answer
33 views

Limit for Brownian local time

If we denote $(B(t))_{t\geq 0}$ as the standard Brownian motion, as by the scaling property, we know for every $n\in\mathbb{N}$, we have $\frac{B(nt)}{\sqrt{n}}\overset{(d)}{=}B(t)$, and denote for ...
Randomwandering's user avatar
1 vote
1 answer
114 views

The function $\log^+x=\max\{1, \log x\}$.

I was reading Marcinkiewicz-Zygmund (MZ) law of large numbers for random fields and came across necessary and sufficient condition $E(|X|\log^+|X|)< \infty$ for MZ-SSLN to hold true. I have a ...
Scion's user avatar
  • 31
1 vote
0 answers
53 views

Interpretation of Spitzer's Law

In the theory of Brownian motion, Spitzer's Law states that: $$\lim_{t\rightarrow\infty}\mathbb{P}\bigg\{\frac{2\theta(t)}{\log(t)}\leq x\bigg\} = \int_{-\infty}^x\frac{1}{\pi (1+y^2)}dy$$ where $\...
EzBots's user avatar
  • 303
1 vote
1 answer
44 views

Two coins, each with P(Head)$\to 0$ as $n\to\infty,$ where $n=n-$th coin toss. Under what conditions is P(both coins Head) infinite times $<1?$

Suppose we have two coins. The probability that the first coin lands on heads is $p_1(n)$ and depends on $n,$ the $n-$th coin toss of this first coin. Similarly, the probability that the second coin ...
Adam Rubinson's user avatar
3 votes
0 answers
56 views

Improved Measure Concentration for small variance random variables.

Consider the random variable $\delta_1$ with the following probability distribution, where we have: $\mathbb{P}(\delta_1=0) = 1 - 2 \delta$, $\mathbb{P}(\delta_1=1)= \delta$, $\mathbb{P}(\delta_1=-1)= ...
darthsid's user avatar
  • 315
2 votes
0 answers
33 views

Infinite divisibility and Gaussian random variables

I was looking for a simple explanation of why the Gaussian random variable can be the only distribution appearing in the Central limit theorem. From the statement of the Central limit theorem, it is ...
foubw's user avatar
  • 1,054
0 votes
1 answer
29 views

limit of convergane in probability [closed]

I was learning about the convergence in probability. I'm unsure if looking at the epsiolon-delta definitions below which captures better the convergence in probability correctly. $$ \forall \epsilon,\...
Tomer Gigi's user avatar
2 votes
1 answer
30 views

Limiting distribution of harmonic sample means of i.i.d. $U(0,1)$s after appropriate scaling.

Suppose $\{X_n\}$ is a sequence of i.i.d. random variables which follow the uniform distribution on $(0,1)$. Denote $$M_n=\frac{X_1+\cdots+X_n}{n},\;H_n=\frac{n}{\frac{1}{X_1}+\cdots+\frac{1}{X_n}}.$$ ...
INvisibLE's user avatar
  • 188
1 vote
0 answers
60 views

Almost sure convergence of $\sum\limits_{1 \leq i < j \leq n} X_i X_j$

I'm trying to prove that: Given a sequence $(X_n)_{n \geq 1}$ of independent and identically distributed random variables, $E(X_i^2) < +\infty$ for all $i \geq 1$, then $$\frac{2}{n(n-1)}\sum\...
Ta Thanh Dinh's user avatar
1 vote
1 answer
43 views

Convergence of positive random vector

Suppose I have a sequence of positive random vectors $\vec{X}_N$ of fixed length $l$. That is, $\vec{X}_N = (x_N^{(1)}, x_N^{(2)},\cdots, x_N^{(l)})$ where each entry $x_N^{(i)} > 0$. Suppose I ...
random_passerby's user avatar
7 votes
1 answer
162 views

Weak convergence of Hilbert-space valued stochastic process

Suppose I have a sequence of stochastic processes $(X_n(t))_{t\in [0,1]}$ such that $X_n(t) \in H$, where $H$ is some separable Hilbert space and $X_n \in C([0,1,], H)$. Goal. I want to show that $...
Nik Quine's user avatar
  • 583
1 vote
2 answers
162 views

Bound on expected norm of the difference between the sample mean $\bar{X_n}$ and population mean $\mu$ as a function of the sample size $n$ for LLN?

My question is motivated by this question: Does law of large numbers converge in $L^1$? that asks about the the convergence in $L^1$-norm of the sample mean $\bar{X_n}$ to the population mean $\mu.$ I ...
Learning Math's user avatar
6 votes
2 answers
544 views

Klenke's proof of Slutzky's Theorem

In Klenke's book on probability he states Slutzky's theorem as: Let $X, X_1, X_2, \ldots$ and $Y_1, Y_2\ldots$ be random varaibles with values in $E$. Assume $X_n \xrightarrow{\mathcal{D}} X$ and $d(...
CBBAM's user avatar
  • 6,277
-1 votes
1 answer
46 views

Proving convergence in prob of $X_n = X = Y_n$ using Markov's inequality.

Below question is from the book 'Probability course.com'. The book provides a solution using Chebyshev's inequality. Before reading that solution, I used Markov's inequality. Is my solution correct? ...
azxz's user avatar
  • 53
1 vote
0 answers
34 views

Random bounded triangular $T_n$ s.t. $Var[T_nS_nT_n^\top]\to 0$ for nonrandom psd $S_n$. Does $(T_n-\bar T_n)S_n\to^P0$ for some nonradom $\bar T_n$?

Let $T_n$ be a sequence of square random matrices with $T_n$ lower triangular with $diag(T_n)=(1,1...,1)$ and $S_n$ a sequence of deterministic symmetric psd matrices. All matrices are in $R^{d\times ...
jlewk's user avatar
  • 2,072
1 vote
1 answer
104 views

What is the probability a random integer $x$ when divided by $3$ has a remainder smaller than when $x$ is divided by $9$? without monte-carlo.

I noticed the quantity of numbers from 1-100 with remainder zero modulo nine = quantity of numbers from 1-100 with remainder one modulo nine > quantity of numbers from 1-100 with remainder 2 modulo ...
user avatar
1 vote
1 answer
52 views

Large Deviation Inequality with a Gaussian type bound

I am trying to follow Terence Tao's notes on concentration of measure, particularly the derivation of equation $(8)$ . Suppose $\{X_i\}$ is a collection of random variables normalised to have mean ...
J.B.R's user avatar
  • 117
0 votes
0 answers
60 views

Do quantiles of quantiles converge to quantiles?

Is there any truth to the statement "the median of medians converges to the true median?". While it's false that the median of medians is the median, is there a way to make this true ...
user125763's user avatar
3 votes
0 answers
82 views

Uniform law of large numbers for ergodic stationary sequence

I am trying to apply a uniform law of large numbers, which is stated in Lemma 7.2 of "Econometrics" by Fumio Hayashi. The starting point is the stochastic process $\{x_t\}$, which we assume ...
Kristan's user avatar
  • 81
8 votes
0 answers
128 views

Question about Ornstein–Uhlenbeck process

Suppose $X(t) = e^{-t/2}W(e^{t})$ is an Ornstein–Uhlenbeck process ($W(t)$ is a Wiener process) and $\epsilon > 0$. I'm trying to find a constant $\delta>0$ so that $$ \lim_T\mathbb{P}\Big( \...
mr_snazzly's user avatar

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