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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19 views

Question about random variables convergence

Further going through old lecture notes I've stumbled upon this... Let's say we are dealing with a sequence of random variables $\{X_n\}_{n=1}^\infty$ such that $\sqrt{n}(X_n-1)\to N(0,2)$ in ...
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20 views

Confidence Interval based on limiting distribution of the Maximum Likelihood Estimator Sigma and Sample Variance

Another statistics question I need some help with. Now, I am not too familiar with limiting distributions (had my course on it over a year ago). I have tried some things myself and ran some ...
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2answers
29 views

Weak convergence result in Levy's Continuity Theorem

I quote a part of Levy's Continuity Theorem and its proof. Theorem Let $\left(\mu_n\right)_{n\geq1}$ be a sequence of probability measures on $\mathbb{R}^d$, and let $\left(\hat{\mu}_n\right)...
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1answer
29 views

Does the convergence in probability imply the following limit is $1?$

Let $X_n \in \mathbb{R}$ be a sequence of non-constant random variable with continuous PDF converging in probability to $c,$ but not necessarily convergence almost surely, i.e. $$\lim\limits_{n \to \...
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1answer
29 views

If densities converge then the corresponding RV converge in distribution

I tried to prove the following Theorem: Given $(X_n)_{n\in\mathbb{N}}$ iid. random variables with $\mathbb{E}[X_i^2]<\infty$. If the rv's have respective densities $(f_n)_{n\in\mathbb{N}}$ and $...
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1answer
45 views

Sum of bernoullis

Let $(X_i)_{i\geq1}$ be $i.i.d$ Bernoulli ($\lambda$) and $S_n = \sum_{i=1}^nX_i $. Find the limiting distribution of $S_n$ as $n \rightarrow \infty$. My approach: Use characteristic functions. $$\...
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1answer
44 views

An application of Lebesgue's Dominated Convergence Theorem. Is that correct?

Let's define a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and consider a nonnegative random variable $Y$ defined on it such that $Y\in\mathcal{L}^{1}$. Since, for a constant $c$, it ...
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1answer
22 views

Is this limiting total variation result valid?

Consider a triangular array of rowise independent random variables $(X_{n,1}, \ldots,X_{n,n})_{n\geq 1}$ and assume $\mu_n=f_n(X_{n,1},\ldots,X_{n,n})$, for some measurable map $f_n:\mathbb{R}^n \to \...
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1answer
16 views

Applying the Central Limit Theorem to show $\lim_{N \rightarrow \infty} \mathbb{P}(0 \leq \sqrt{12N}(\overline{X}_n -\frac{1}{2}) \leq 1) = I$

Consider $(X_n)_{n \in \mathbb{N}}$ of i.i.d. random variables with $X_n \sim \text{Uniform}(0,1)$. We set $$\overline{X}_n = \frac{1}{N}\sum_{i=1}^N X_i$$ and we calculate $\mathbb{E}(\overline{X}_n)...
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44 views

convergence a.s. and in probability of discrete random variable

Assume $X_{n}$ is a sequence of random variables taking values from some finite discrete space. Next, assume $$ P(X_{n} = c) \to 1, $$ for some $c$, as $n \to \infty$. I would like to find an ...
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31 views

$X_p \sim GEO(p)$. What's the limiting distribution of $X_p$ as $p \to 0+$? (Should be $EXP(1)$)

I've been given the above question, and I've tried getting $EXP(1)$, but no matter what I try fails. We've learned about Characteristic Functions, so I assume we need to show that the $$\lim_{p \to ...
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24 views

Applying de Moivre-Laplace Theorem correctly

Assume $10\%$ of the population are left-handed. We consider a stadium with $5000$ people. What is the probability that there are between $750$ and $800$ left-handers in the stadium? So we have $p = 0....
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3answers
41 views

Limit of Binomial CDF

Let $\operatorname{Binom}(n, p)$ be $n$ trials with probability of success $p$. I want to find $$\lim\limits_{n \to \infty} P(\operatorname{Binom}(n, p) \geq n/2)$$ I didn't know how to do this with ...
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1answer
34 views

Show that maximum of iid. random variables converge in mean

I do an exercise from a textbook, where one has to show for $(X_n)_{n\in\mathbb{N}}$ iid. having finite second moment that $$\frac{1}{\sqrt{n}}\max_{1\le i\le n}|X_i|\rightarrow 0$$ in $L^1$. For this ...
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Understanding stochastic Big-$O$ notation in a stochastic process context

I am trying to understand an equality in this paper about locally stationary processes on page 24. The equality includes stochastic Big-O (see D5 in this paper for a definition of stochastic O-...
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39 views

What are sufficient conditions such that consistency of ML estimate implies consistency of MAP estimate?

I am interested in under what conditions the frequentist consistency of a Maximum-Likelihood estimator is enough to give the consistency of a maximum-a-posteriori point estimate, with the further ...
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20 views

A sufficient condition for almost sure convergence.

I have been working on the following problem. Prove that if a sequence $X_n$ of random variables satisfies $$\lim_{n,m\rightarrow \infty}P\left\{ \sup_{m<k\leq n} |X_k-X_m|\geq \delta \right\...
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Where can I find the derivation of the formula used here?

Brian Tung's answer to this question is stated as follow: $$0.85n + 2.33(0.357\sqrt{n}) \leq 144$$ Which is seems to be: $$\mathbf E[X] + z\cdot\sigma_X \leq \mu$$ I would like to ask where I can ...
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1answer
65 views

Concentration of norm of linearly transformed normal random vector as dimension go to infinity

Following no response, recently asked on MO. Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $Bx_k:= x_k/ {k}, 1 \...
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2answers
115 views

Conditions for a.s convergence of a gamma series

Let $(X_n)$ be a sequence of independent random variable, $X_n$ having the gamma distribution, $$f_{X_n}(x)=\frac{\alpha_n^{p_n}}{\Gamma(p_n)}e^{-x\alpha_n}x^{p_n-1}1_{[0,+\infty[}(x).$$ Find ...
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2answers
24 views

Bounded probability implies convergence in probability

Let $(X_n)$ be a sequence of random variables and $(a_n),(b_n)$ be two sequences of non-negative real numbers such that $a_n\downarrow 0$ and $b_n\downarrow 0$ when $n\to\infty$. If for any $t>0$,...
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1answer
50 views

Determining norming and centering constants to guarantee weak convergence to a non-degenerate distribution

Problem Setting. Suppose $\{X_n\}_{n\in\mathbb{N}}$ are independent, identically distributed random variables with mean $m$ and variance $\sigma^2$. Consider the sequence of random variables $\{Y_n\}...
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1answer
43 views

distribution convergence N(0,σ^2/n)

Consider the sequence $X_1, X_2 \dots$ where each $X_n$ has a normal distribution $\mathcal{N}(0, σ ^ 2 / n)$. Let us show that $X_n$ converges in distribution to 0. $$F_{X_n} (x) = \frac{1}{\sqrt{2\...
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1answer
30 views

Beginner limits in Calc

I got the first part correct(graph) but for the second, is it asking for the values that make y=0? Im confused on this part
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41 views

Convergence in distribution and in probabiliity

Let $(X_n)_n$ be a sequence of independent random variables. Let $Y_n=\sum_n X_n.$ Suppose that $(Y_n)_n$ doesn't converges in probability then we can find $\epsilon>0,$ sequences $(p_n)_n$ and $(...
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45 views

Please verify my proof for the lemma: “If $X_n \to_d X$ and $X_n + Y_n \to_d X$ with $X_n$ and $Y_n$ independent for each $n$, then $Y_n \to_p 0$”

It is the lemma 5.1 in this paper: https://dornsife.usc.edu/assets/sites/1193/docs/lin.pdf, and the paper contains the proof. My attempt is: By Portmanteau theorem, $X_n \to_d X$ is the equivalent ...
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23 views

Diagonal convergence of empirical measures of empirical measures

Given a probability measure $\mu$ (with density if needed), and its empirical measures $(\mu_n)_n$ which converges to $\mu$ in distribution almost surely as $n\to \infty$, we know: for fixed $n$, the ...
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1answer
92 views

Using Lyapunov Condition show that a sequence of Poisson-Binomial random variables converge in distribution to the standard normal.

Let $B_{k}$, $k \geq 1$ be independent Bernoulli random variables such that $P(B_k = 1) = \frac{1}{k} = 1 - P(B_k = 0)$. Note that these are also known as Poisson-Bernoulli random variables. Now, I ...
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1answer
100 views

Detail on Portmanteau theorem

Let $(\mu_n)_n$ and $\mu$ be probability measures on $(\mathbb{R}^d,B(\mathbb{R}^d))$. In Portmanteau theorem, one can prove that $(\mu_n)_n$ converges weakly to $\mu$ if and only if for all bounded, ...
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1answer
29 views

Rigorous definition of convergence in mean

I'm looking at the wikipedia definition for Convergence in Mean, and I would appreciate clarification on one part. It says, given a real number $r \ge 1$, we say that the sequence $X_n$ converges in ...
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1answer
33 views

Convergence to $+\infty$ a.s along a subsequence

We say that $(X_n)_n$ converges in probability to $+\infty$ if $$\forall \epsilon>0,\lim_n P(X_n<\epsilon)=0.$$ Prove that there exist a subsequence $(X_{\phi(n)})_n$ which converges to $+\...
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57 views

Central Limit Theorem for Linear Combination of Poisson Variables

Let $ X_1,X_2,...,X_N $ be independent Poisson variables with different parameters $\lambda_n$ that potentially can be quite small (e.g., $10^{-4}$). I am interested in the distribution of $Y$ which ...
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3answers
164 views

Weak convergence of a sequence of gaussian random vector

Let $(X_n)_n$ be a gaussian random vector taking values in $\mathbb{R}^d,$ let $K_{X_n}$ denote the covariance matrix of $X_n.$ Show that if $(X_n)_n$ converges in distribution to $X,$ then $(K_{X_n})...
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2answers
183 views

Random equation

We want to prove the following proposition: Let $X$ and $Y$ be two random variables independent and identically distributed with variance $\sigma^2.$ Let $\alpha,\beta \in \mathbb{R}$ such that $\...
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1answer
34 views

$\delta$-method on $\mathbb{R}^d$

Let $(X_n)_n$ be a sequence of i.i.d random variables taking values in $\mathbb{R}^d,d \geq 1,$ such that $E[||X||^2]<+\infty$ where $||.||$ is a norm on $\mathbb{R}^d,$ and let $\overline{X}_n=\...
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1answer
22 views

What integral theorem can justify this limit of probabilities, where we have a limit in the integrand and the limits of integration?

Let $(Y_n)$ be a sequence of random variables with finite mean and variance such that $Y_n$ converges in distribution to a random variable $Z$ with finite mean and variance (so that in particular, $P(|...
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1answer
26 views

A sequence of random variables with bounded first three moments

Suppose $X_n$ is a sequence of random variables such that $$\mathbb{E}(X_n)=0, \mathbb{E}(X_n^2)=1, \mathbb{E}(X_n^3)\leq 1$$ Do we have the following result: for any $\epsilon>0$, $$\mathbb{E}(X_n^...
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1answer
56 views

Strong law of large numbers for triangular arrays

Consider a triangular array $X_{n,1},\ldots,X_{n,n}$ of rowwise i.i.d. real random variables with $ \sup_{n \in \mathbb{N}} \mathbb{E}\vert X_{n,1} \vert < \infty$ and $ \lim_{n \rightarrow \infty}...
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41 views

An application of the convergence of types theorem

Let $(X_n)_n$ be a sequence of random variables such that there exist $u_n>0$ and $p_n \in \mathbb{R},$ such that $Y_n=\frac{X_n-p_n}{u_n}$ converges in distribution to a non-degenerate random ...
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1answer
144 views

Weak law of large numbers for dampened Cauchy distribution

Say I have a sequence of real independent functions $X_i$ on a probability space $(\Omega, P)$ such that $(X_i)_* P$ gives the probability distribution $d\alpha := \frac{cdx}{(1+x^2)\log(1+x^2)}$ (say ...
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1answer
50 views

How do I determine $\lim_{n\to \infty}P(|\sum_{i=1}^{n^2}X_i - n^2|\leq n^{\alpha})$ where $(x_i)$ is a sequence of random variables? [closed]

Let $x_1, x_2, \dotsb$ be a sequence of independent random variables distributed uniformly on the interval $[0, 2]$. Find all possible values of the following limit, where $\alpha$ is a positive ...
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1answer
14 views

Convergence of the product of two Random Functions

$lim_{t\to\infty}\mathbb{P}(|g(X(t))|>\epsilon)=0$, $f(X(t))=\mathbb{1}_{({X(t)}\neq{0})}$ I want to show $f(X(t))g(X(t))$ coverage in probability to 0. Just wondering if it is possible, thank ...
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1answer
166 views

If a probability density function has a limit at infinity, is that limit always zero? [duplicate]

I just read a related question which showed that probability density functions do not always have a limit of zero as a variable approaches infinity because they don't always have a limit at infinity. ...
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22 views

Argmax of functions of posterior under bernstein-von-mises theorem

Im reading Asymptotic Statistics (Van der Vaart, A. W. (2000)), where in chapter 10 the Bernstein-von-Mises theorem is discussed. So assume we are in the situation where $||P_{(\Theta|X_1,...X_n)} - ...
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11 views

Converting a trimodal distribution into a uniform distribution

Consider some generator $G$ that outputs discrete points trimodally distributed along the interval $(0, 1)$, with the points tending to cluster around 0, 1, and 0.5. Then, I need to uniformly ...
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1answer
51 views

Weak convergence of $\mathcal{U}([u_n,v_n])$

If $(u_n)_n$ and $(v_n)_n$ are two sequences of real numbers, such that $\forall n \in \mathbb{N},u_n<v_n,$ if $X_n$ is a random variable with density $f_{X_n}=\frac{1}{v_n-u_n}1_{[u_n,v_n]},$ and ...
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1answer
42 views

Irreducible Markov chain with µ as the stationary distribution.

Prove that if $\mu$ is a stationary distribution for an irreducible Markov chain on $S$, then $\mu(j) > 0 \ \forall j \in S$. (only using the fact that Markov chain is irreducible and that $\mu P = ...
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61 views

Big O under the integral / taylor expansion of likelilhood

Assume we have some fixed (finite, may be a density, but not necessarily) function $f(x)$ and some other function $g(\theta,x)$ such that for fixed $x$, $g(\theta,x)$ is in $O(\theta^m)$. ...
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2answers
27 views

Limiting distribution to Weibull

I am trying to solve the following question: Let $X_1,X_2,...$ i.i.d with $F(x) = P(X_i < x),~\\M_n := \max_{1\le{i}\le{n}}\ X_i,~\ $ $F(x_0) = 1$ and $F(x)<1$ for all x Given; $\lim_{x\...
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19 views

Is there an inequality relating the difference between two random variable and the difference between their distribution functions?

Let $X, Y: \Omega \to \mathbb{R}$ be two random variables with respective cumulative distribution functions $F_X, F_Y$ respectively. I was wondering whether there's an inequality possible relating the ...

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