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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Speed of convergence of empirical probability mass function

I was wondering about the following situation: Suppose we have a random variable $X$ taking values in the finite set $\{1,2,\ldots,k\}$. Let the probability mass function be denoted by $f$. Suppose we ...
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Central limit theorems and almost sure invariance principles

this is a more general question. Consider a sequence $(X_j)_{j \in \mathbb{Z}}$ of iid real-valued random variables with mean zero and $\mathbb{E}(X_1^2) = 1$ on a probability space $(\Omega, \mathcal{...
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Covariance matrix and the Cramer-Wold device

In Appendix B of Financial Statistics and Mathematical Finance, after the Cramer-Wold device (Theorem B.1.1) it is said: In particular, the Cramer–Wold technique tells us that $$X_n \xrightarrow{D} N(...
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Tail bounds for sub-Gaussian and sub-exponential distributions

Massart and Laurent (see [1], Lemma 1 on page 1325] give tail bounds for $\chi^2$ random variables. A corollary of their bound is the following: $$P\left[\frac{1}{k} X \leq 1- 2\sqrt{\frac{x}{k}} \...
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Return of Brownian motion to zero

From chapter 4 of Bulinskiy & Shiryaev's Theory of Random Processes (ISBN 5-9221-0335-0): [Exercise 26] Let $W = \{W_t, t \geqslant 0 \}$ be a $m$-dimensional Brownian motion, where $m \geqslant ...
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Concentration of probability measures

Assume we have a vector $X$ consisting out of $N$ independent random variables $X_1, ..., X_N$. We determine ("measure") the value of "k" (say, for simplicity, the first $k$) of ...
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Does the infinite limit probability of additive closure for pairs of elements from $(0, 1, 2 \cdots, n)$ not exist?

Suppose that we have a sequence of non-negative integers $S_n = (0, 1, 2 \cdots, n)$. At a given value of $n$ we can uniformly sample two elements of $S$ with replacement. We can check if adding those ...
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Why is this estimator with ill-defined moments useful? And why is the Cauchy PV of its expectation integral a reasonable measure of center?

This question pertains to the paper (available online through JSTOR): M. H. QUENOUILLE, NOTES ON BIAS IN ESTIMATION, Biometrika, Volume 43, Issue 3-4, December 1956, Pages 353–360, https://doi.org/10....
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Let $Y_n := \frac{X_1\xi_1 + \cdots + X_n \xi_n}{\xi_1 + \cdots + \xi_n}$. Is $\operatorname{var} Y_n \to 0$ as $n \to \infty$?

Let $X, X_1, X_2,\ldots$ be an i.i.d. sequence of random variables with finite variance. Then $$ \operatorname{var} \left ( \frac{X_1 +\cdots + X_n}{n} \right ) = \frac{\operatorname{var} X}{n} \to 0 \...
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Limit of a Parametrized Expectation

$X$ is a non-negative random variable and $\beta$ is a real-number parameter. I am interested in finding whether the limit $\lim_{\beta\rightarrow 0} \left( E[X^{\beta}] \right)^{\frac{1}{\beta}}$ ...
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Proving convergence in distribution of quotient of two random variables

Question. Let $X_{1}, X_{2}, \ldots$ be positive i.i.d. random variables with $\mathbf{E} X_{1}=a$ and $\operatorname{Var}\left(X_{1}\right)=\sigma^{2}$. Let $S_{n}=X_{1}+\cdots+X_{n}$. Show that $Y_n ...
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Prove that $\lim_{n\rightarrow \infty} \mathbb{E}(YX_n) = 0$ for uncorrelated $(X_n)$ with $0$ mean, $\rm{Var}(X_i)=1, Cov(X_i,X_j)=0$ and bounded $Y$

Let $(X_n)$ be a sequence of uncorrelated random variables (i.e., $\operatorname{Cov}(X_i,X_j) = 0$ for i , j) with expectation $0$ and variance $1$. Prove that for any bounded random variable $Y$, we ...
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Probability of finding a specific colored plant in a row.

In a packet of flower seeds 3/4 are yellow flowering and rest are white. If 200 rows of each plants are planted, how many will contain (i) all yellow flowers. (ii) all white flowers (iii) two yellow ...
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Convergence in probability implies a.s. convergence in a countable space

Let be $(\Omega, \mathcal{F}, \mathcal{P})$ a probability space, $(X_{n})$ a sequence of randiom variables and $X$ a random variable. Let be $\Omega$ countable and $\mathcal{F}$ a power set of $\Omega$...
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Limit of the smallest eigenvalue of a random Gram matrix $XX^T/n$.

Let $X\in\mathbb{R}^{n\times d}$ be a random matrix with independent rows which satisfy that $\mathbb{E}(X_iX_i^T) = I_d$ and $\mathbb{E}(X_i) = 0$. The results of P. Yaskov provide conditions on when ...
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How to determine the distribution of the limiting random variable

Given a sequence of random variables $\{X_n\}_{n \geq 0}$ defined as follows. $X_0 = p, X_{n+1} = qX_n + (1-q)1_{Y_{n+1} \leq X_n}, \forall n \geq 0, $ where $p, q \in (0, 1)$ are constants and $Y_i \...
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(convergence and probability) If $X_n$converges on the probability in $X$. Prove...

If $X_n$ converges on the probability in $X$. Prove a) (using only the definition of convergence with probability) For every $ \epsilon_k \to 0$ when $k \to \infty$, that there exists a $n_k$ such ...
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(strong law of large numbers) We played a game in a casino. $X_i$ the money we won or lost the i-th time....

>We played a game in a casino. $X_i$ the money we won or lost the i-th time. Each time that we win, we take 1 dollar. When we lost, we lost 1 dollar. If p is the probability of winning and q the ...
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Theorem about trimmed mean and location parameter, explanation needed.

The theorem is found on Stephen M. Stigler paper The Asymptotic Distribution of The Trimmed Mean (1973) Let $S_n$ denote trimmed mean and: $$a = \sup\{x: F(x) \le \alpha\} \\ b = \inf\{x:F(x)\ge \beta\...
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Increment property of Brownian Motion

I'm trying to prove an increment property for the Brownian motion, but I'm unable to figure it out, maybe someone can help me out. Consider two sequences $(a_n)_{n \in \mathbb{N}}$, $(b_n)_{n \in \...
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If a random matrix converges to an invertible constant matrix $A$, does its inverse converge to $inv(A)$?

Let $B_n$ be a sequence of random matrices and $$ \underset{n\rightarrow \infty}{\text{plim}}(B_n) = A, $$ with A invertible. Does this imply that $$ \underset{n\rightarrow \infty}{\text{plim}}(B_n^{-...
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Probability of zero from infinite limit of Laplace transform

Can someone remind me why, for a non-negative random variable $X$, this equality holds: $$\mathbb{P}[X=0]=\lim_{\lambda\rightarrow\infty}\mathbb{E}\left[e^{-\lambda X}\right]$$
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1 answer
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Converse Version of Slutsky's Theorem

Let $X_n$ and $Y_n$ be two random sequences. According to Slutsky's theorem, we know that if $X_n \overset{P}{\to} c$ for $c \in \mathbb{R}^1$ and $Y_n \overset{d}{\to} Y$, then $X_nY_n \overset{d}{\...
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Find transition matrix for Markov Chain

The problem is the following: An individual has three umbrellas, some at her office, and some at home. If she is leaving home in the morning (or leaving work at night) and it is raining, she will ...
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Consistency of Scaled Sample Variance

Let $X_1, X_2, \dots, X_n$ be iid random variables. Consider the sample variance $$S_n^2:=\frac{1}{n-1} \sum_{i=1}^{n}(X_i - \overline{X})^2$$ where $\overline{X}:=\frac{1}{n}\sum_{i=1}^n X_i$ is the ...
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Doubt about Durrett's Multivariable Limiting Distribution Example

I am trying to understand how limit theorems work in multivariable scenarios, but I am having some difficulty understanding an example (Example 3.10.9 of Durrett). The multivariable CLT says: Let $X_1,...
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distribution of lengths for two cycles in constrained random mapping

Let $f$ be a uniform random endomorphism on $\{1,2,...,n\}$. We say $f$ is connected if its functional graph possesses exactly one connected component. The unique cycle in this component has length $...
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About minimum 2 and maximum 9 probability

There are 33 students in a class. What is the probability that at least two of the 9 randomly selected people were born in the same month? I know classical birthday problem ( 1-(365/365.)(364/365)...) ...
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1 answer
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Generalizing Bernoulli's theorem to any sequence of IID random variables

Bernoulli's Theorem says the following: If S represents the number of successes obtained during $n$ Bernoulli trials, and if $p$ is the probability of success, then $$\dfrac{S}{n} \to p \text{ as } n \...
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1 vote
1 answer
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Does a Gaussian random walk lead to a Gaussian distribution in the limit, even when the initial state is non-Gaussian?

Suppose we have an initial random variable, which is not Gaussian, but has mean $0$, std $1$. Now we add $N$ unit Gaussian variables to this initial random variable, and then renormalize to mean $0$, ...
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2 votes
1 answer
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Limit of expectation of maximum of standard normal distribution

I cannot find the way to show that if $X_1,\dots,X_n$ are independent standard normal random variables, then $$ \lim_{n \rightarrow \infty} \frac {\mathbb{E} \max_{i=1,\dots,n}X_i}{\sqrt{2\log n}} = 1 ...
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1 vote
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Estimate of Expectation of Estimated Function

Let $X_1,X_2,...$ be a sequence of random values distributed uniformly in $[0,1]$. By the law of large numbers, we have for any integrable function $f=f(x)$ (convergence in probability to an ...
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1 answer
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The order statistics of uniform distribution $U_{n,{k_n}}$ times $n$ tends to $\infty$ in probability.

Let $U_1,U_2,...U_n,$ be iid samples from uniform distribution $U(0,1)$. And the order Statistic: \begin{equation} U_{n,1}\leq U_{n,2}\leq ...\leq U_{n,n}, \end{equation} Let $1\leq k_n \leq n$ and $...
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Estimate the sample size in proportional stratified random sampling to get the same precision of simple random sampling

I was reading example 3.2 on stratified sampling in Sampling: Design and Analysis by Lohr where a stratified random sampling design is proposed and compared with a simple random sampling design : The ...
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2 votes
1 answer
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Durrett's Continuity Theorem Proof (Theorem 3.3.17)

I am trying to understand Durrett's proof of the continuity theorem. It contains two parts. The one I have problems with is the second part, whose statement is: If $\varphi_n(t)$ converges pointwise ...
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4 votes
1 answer
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The order statistics of uniform distribution $U_{n,k_n}\rightarrow p$ a.s., when $\frac{K_n}{n} \rightarrow p$

Let $U_1,U_2,...U_n,$ be iid samples from uniform distribution $U(0,1)$. And the order Statistic: \begin{equation} U_{n,1}\leq U_{n,2}\leq ...\leq U_{n,n}, \end{equation} Given $p\in(0,1)$, if $1\leq ...
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1 answer
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Why is $\frac{1}{n} \sup_{k\le n} |T_k - k| \to_{a.s.} 0$ if $T_n/n \to E T_1 = 1$?

Let $T_n$ be a sequence $0\equiv T_0 \le T_1 \le T_2 \le \cdots$ of random variables such that $T_n - T_{n-1}$ is distributed to a zero mean unit variance random walk so that $E T_n = n$. Then by the ...
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3 votes
1 answer
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If $X_n \overset{\text{a.s.}}{\to} \infty$ and $Y_n = O_P(1)$, does $X_n + Y_n \to \infty$ in probability or almost surely?

Question Let $(X_n)$ and $(Y_n)$ be random variables on the probability space $(\Omega, \mathcal A, P)$. Suppose $X_n \overset{\text{a.s.}}{\longrightarrow} \infty$ and $Y_n = O_P(1)$, i.e. $(Y_n)$ is ...
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1 vote
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Big $\mathcal{O}$ of a Tuple

I have seen the following variant of big-O notation written in a textbook (page 1 of Barbour, Holst, and Janson): $f(m) = \mathcal{O}(g(m), h(m))$ as $m \to \infty.$ Is this notation standard? What ...
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3 votes
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Generalization of central limit theorem to sums of the form $\sum_{k=1}^n\frac{1}{X_k^m}$, $m=1,2,\dots$?

Consider this lesser-known fact: If $X\sim\mathcal N(\mu,\sigma^2)$ then $$ \frac{\frac{1}{n}\sum_{k=1}^n\frac{1}{X_k}-\mathsf E_\mathcal PX^{-1}}{\pi f_X(0)}\overset{d}{\to}\operatorname{Cauchy}(0,1)...
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Why is convergence of the average of variances guaranteed for independent random variables with bounded moments?

I'm trying to understand a proof of the central limit theorem for independent random variables that satisfy a condition of bounded moments, but there is a step I do not get. Let $(a_i)^\infty_{i=1}$ ...
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Asymptotic convergence of shifted and scaled Wigner matrix

Let $\frac{X}{\sqrt n}$ be a Wigner matrix, such that $X_{ij}$ are iid random variables, with mean 0 and variance 1, with $X_{ij} = \overline{X_{ij}}$ for $i > j$. Then we know by Wigner's Theorem ...
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2 votes
1 answer
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Convergence in Kolmogorov distance

Sequence of random variables $\left\{X_n\right\}_{n=1}^\infty$ converge to random variable $X$ in Kolmogorov distance if $$\lim\limits_{n\to \infty}\left(\sup\limits_{x\in\mathbb{R}}|F_n(x)-G(x)|\...
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  • 519
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0 answers
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A new convergence problem for the conditional expectation

You have risks $X_1$, $X_2$, ... (they are assumed to be independent, but not necessarily identically distributed) and $S_n= X_1 + X_2 + \cdots +X_n$ QUESTION: under what reasonable conditions we have ...
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1 vote
0 answers
45 views

Why is $ \mathbb{E} \limsup\limits_{n\rightarrow\infty}X_n = \mathbb{E} \lim\limits_{n\rightarrow \infty} X_n$

For a sequence of positive random variables $X_n$ (which i assumed is not nessesarily monotonically non decreasing). Assuming that $\lim\limits_{n\rightarrow \infty} X_n = X$ I don't understand why $ \...
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  • 519
0 votes
1 answer
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For i.i.d sequence of random vectors $(X_i,Y_i), i=1,...,n)$ show that $\sqrt{n}(\bar{XY}- \bar X\bar Y -E(XY)+E(X)E(Y)) \to_d N(0,\alpha^2)$

For i.i.d sequence of random vectors $(X_i,Y_i), i=1,...,n)$ show that $\sqrt{n}(\bar{XY}- \bar X\bar Y -E(XY)+E(X)E(Y)) \to_d N(0,\alpha^2)$$, assume all moments are finite. Also find an expression ...
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2 votes
1 answer
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Convergence in probability and convergence in distribution in the context of OLS

What I am missing here? Let there be three iid variables $(W_i, D_i, U_i)$ where $U_i$ may have an error interpretation in a traditional OLS context. That is $E[W_iU_i]=0$ and $E[D_iU_i]=0$. Moreover, ...
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2 votes
0 answers
72 views

A question regarding the central limit theorem

I found the following exercise on an old exercise sheet of my university and I have some problems with understanding the result: Let $X_1\dots \ X_n$ be iid random variables with $X_1\sim$ $Ber(p)$ ...
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2 votes
0 answers
30 views

Property of sums of independent and identically distributed random variables

Let $X_i(i=1,...,n)$ be independent and identically distributed random variables whose probability density functions (PDF) are $f(x)$. Known that $f(x)$ is a continuous even function defined within $\...
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Calculating probabilities from multiple flux distributions

I'm trying to solve a physics problem. I have three flux functions that quantify the frequency (or flux) of incoming particles according to some property of the particles. The first frequency is a ...
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