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.
1,489
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Gambler's ruin in the limit (only stopping rule ruin)
Imagine a classical Gambler's ruin with winning probability p and losing probability q = 1-p. You start at 1\$ and lose once you reach 0\$. The only stopping rule is that the game is over when the ...
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Does this simple branching random walk on $\mathbb Z$ satisfy a central limit theorem?
Consider the following simple branching random walk on $\mathbb Z$ in discrete time:
At stage 0, we start with one token at the site 0. At each time step, we randomly, independently split each token ...
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Is the random limit in the Kesten–Stigum theorem random in this simple case?
Fix $p \in (0,1)$ and suppose we have a sequence of random variables defined as follows: let $X_0 = 1$, and given $X_n$, we have the binomial distribution
$$X_{n+1} \sim \text{Bin}(X_n,p) + X_n.$$
I.e....
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Strong consistency of kernel density estimator
I am studying the book Nonparametric and Semiparametric Models written by Wolfgang Hardle and have difficulty with the following exercise:
$\textbf{Exercise 3.13}$ Show that $\hat{f_h}^{(n)}(x) \...
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Expectation of $\left| X - a \right|$ and the generator of Hermite polynomials
Suppose we have a random variable $X$ with an unknown distribution, but known first $N$ moments $\left\{ E [ X^n] \right\}_{n=1}^N$. We are asked to calculate the value of
\begin{equation}
E \left[ | ...
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$\Pr(X_n\leq x) \to \Pr(X\leq x)$ for $x\geq 0$ and $\Pr(X_n< x) \to \Pr(X\leq x)$ for $x<0$ imply convergence in distribution?
Let $X_n$ be random variables and let $X\sim \mathcal{N}(0,1)$ such that $$\forall x\geq 0, \quad \Pr(X_n\leq x) \to \Pr(X\leq x)$$
and
$$\forall x< 0, \quad \Pr(X_n< x) \to \Pr(X\leq x).$$
Does ...
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This is an unsolved interview question. Can anyone solve this? [closed]
You play a game where you need to reach a wall 100 steps ahead of your starting position You roll a six-sided die and walk forward a number of steps based on the number the die lands on. However, if ...
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Generalization of Marchenko-Pastur law [closed]
I have a question about the random matrix theory. Say $X \in R^{M \times N}$ is a random matrix such that the each row of $\sqrt{N} X$ are independently complex random variables with mean zero, ...
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Durrette Exercise 2.2.9 - Weak Law for Positive Variables
I'm working my way through Rick Durrett's book "Probability: Theory and Examples" and want help solving the following question.
$\textbf{Question:}$ Suppose $X_1,X_2,\ldots$ are i.i.d random ...
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Show a $\liminf$ statement involving probabilities
Take a sequence of binary random variables $(Y_t)_t$ such that
$$
(A) \quad \Pr\Big(\lim_{T\rightarrow +\infty} \frac{1}{T}\sum_{t=1}^T Y_t=\nu\Big)=1.
$$
Consider another sequence of discrete random ...
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Law of Large Numbers for Conditioned Mean
Let $\{X_i\}$ be a sequence of i.i.d. random variables, $\overline X_n = \displaystyle\frac{1}{n}\sum_{i=1}^n X_i$ be their $n$th partial mean, and $\mu = \mathbb E[X_1] < \infty$.
Let $x \in \...
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Weak convergence does not imply joint weak convergence?
Suppose that $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ as $n\to\infty$ where "$\Rightarrow$" means convergence in distribution. We know that it does NOT imply that $(X_n,Y_n)\Rightarrow (X,Y)...
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Can weak convergence imply orders of mean and variance? [closed]
Suppose we have the following weak convergence:
$$
n^{-1/2}\cdot (X_n-n) \Rightarrow X \quad \text{as} \quad n\to\infty,
$$
where $X_n$ and $X$ are random variables (note: $X$ is a non-degenerate r.v.)...
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How to prove the following joint weak convergence?
Given a $T>0$, let $\mathcal{C}[0,T]$ be the space of continuous functions on $[0,T]$. Let $Y_n(t)$ be stochastic processes in $\mathcal{C}[0,T]$. We define the weak convergence in the sense of ...
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The difference between the Bayesian estimator and MLE multiplied by $\sqrt{n}$ converges to zero.
Let $\theta$ be a random parameter with support $[0,1]$ and positive density, and let $X_1,X_2,\ldots \sim \rm N(\theta,1)$ be its i.i.d. observations. Define
$$ \delta_n := \sqrt{n} \Big(\hat \...
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Expected squared difference between between the ML estimator and the posterior expectation.
Let $\theta$ be a random parameter with support $[0,1]$ and positive density, and let $X_1,X_2,\ldots ~ \rm N(\theta,1)$ be its i.i.d. observations. Does
$$ \rm E\Big[n\cdot \Big(\hat \theta_n(X_1,\...
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An example of a sequence of RVs converging to 1 a.s., but being negative with probability converging to 1.
Is there a simple example of a sequence $X_1,X_2,\ldots$ of random variables such that $X_n \to 1$ almost surely, but $P(X_n\leq 0) \to 1$.
I'm trying to understand the difference between convergence ...
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Switching integration and minimization for positive random variables? [closed]
Suppose we have three non-negative random variables $X,Y,Z$, they may be dependent and they satisfy $Z\geq\min\{X,Y\}$. Now we know that there exists a constant $A$ such that $E(X)\geq A, E(Y)\geq A$. ...
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Central limit theorem for sum of dependent Bernoulli random variables in multivariate hypergeometric setting
I am struggling to find a suitable Central Limit Theorem (CLT) for dependent variables for the following example:
We have $K$ bins. In each bin $B_i$, $i=1,...,K$, we have $N_i$ balls, with $N=\sum_{i=...
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Almost sure convergence of sum of independent random variables in the book by Hall and Heyde
In Section 1.3 of the book "Martingale Limit Theory and Its Applications" by P. Hall and C. C. Heyde, the authors give a martingale proof for the following result: If $S_n = \sum_{i=1}^n X_i$...
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56
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Support of Bohr-Jessen random variable at 1
I am reading the book by E. Kowalski on Probabilistic Number Theory. Link to the book: https://people.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf
I am stuck on Exercise $3.2.4$. It seems ...
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Generalized CLT for uncorrelated centered RVs?
Recall that the classical Central Limit Theorem states that for any sequence of I.I.D random variables $ X_{i} \sim X $ such that $E(X)=0,Var(X)=1$ we have
$\frac{\sum_{i=1}^{n}X_{i}}{\sqrt{n}} \...
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Longest Common Subsequence (LCS) - Subadditive Ergodic Theorem - Limit Upper Bound
I'm reading Durrett's Probability : Theory and Examples and in the Subadditive Ergodic Theorem Chapter I'm trying to solve the exercise about bounding the limit of the longest common subsequence ...
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Convergence of random vectors to vector with independent marginals when marginal distribution function converges at points of continuity
Let $\xi_n, \xi, \eta_n, \eta$ be random vectors in $\mathbb{R}^{d}$ such that $\xi$ and $\eta$ are independent, $\xi_n \overset{d}{\to} \xi$ and $\eta_n \overset{d}{\to} \eta$. Denote by $C(F_{\xi})$ ...
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On the convergence of the infimum of an average
The strong law of large numbers states that the sample average converges almost surely to the expected value: $\frac{1}{n}\sum_{i=1}^ng(X_i,t)\rightarrow\mathbf{E}[g(X,t)]$ (a.s) for all $t\in\mathbb{...
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Does convergence in probability imply deterministic convergence for non-random sequences?
Suppose we know that an estimator $\hat{\theta}_n$, which is a function of a random sample $X = (X_1, \dots, X_n)$, converges in probabiltiy to some constant $\theta$, i.e.,
$$\forall \varepsilon > ...
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A question related to the convergence of characteristic functions
Assuming the following assumptions:
$\underset{n \to \infty}{\limsup} \sum_{j=1}^n | 1- \varphi_{jn}(u)|< \infty$;
$s_n = \sup_{1\leq j \leq n}|1- \varphi_{jn}(u)| \to 0$, as $n \to \infty$
I ...
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Local limit theorem for not-identically distributed series?
I'm wondering if there is a local limit theorem for sums of independent random variables $X_1+X_2+\dots+X_n$ which are independent but not identically distributed? I know some form of CLT holds with ...
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Convergence of the absolute value of a random series
Let $(X_n)_n$ be a sequence of independent Bernoulli$(p)$ random variables, i.e., $P(X_n = 1) = p$ and $P(X_n = 0) = 1 - p$. Does
$$\lim_{n \to \infty}\left|\frac{1}{n}\sum_{k = 1}^n(-1)^{\sum_{j = 1}^...
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What does Borel's large number theorem really mean?
According to this page in "Encyclopedia in Mathematics", the Borel's large number theorem can be stated as below.
"Consider independent random variables $X_1,\dots,X_n,\dots$ which are ...
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$X_n \to_p X$, how to get sign consistency at rate $a(n)$, i.e $a(n)P(sgn(X_n) \neq sgn(X)) \to 0$
Let $X_n$ be a sequence of random variables which converge in probability to $X$. Suppose also that for all $\varepsilon > 0$ there exists $\delta_\varepsilon > 0$ such that $P(|X| \leq \delta_\...
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Can a sequence $X_n$ of positive random variables converge to infinity almost surely with expectations remaining bounded?
Is there some probability space $ (Ω, \mathcal F, \mathbb P)$ and a sequence $(X_n)$ of positive random variables on $Ω$ for which $X_n → ∞ $ almost surely, but the sequence $\mathbb E[X_n]$ of ...
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From almost surely to expectation
Suppose you have two sequences of positive random variables, $(X_n)$ and $(Y_n)$, such that a.s. $Y_n = o(X_n)$. Do you see any sufficient condition to obtain $\mathbb E(Y_n)=o(\mathbb E(X_n))$?
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$X_n\to X$ and $Y_n\to Y$ in distribution. What we can say about the convergence of $X_n+Y_n$?
Suppose we have two sequence of random variables $X_n$ and $Y_n$ such that they converge in distribution respectively to $X\sim\Gamma(\frac{1}{2},1)$ and $Y\sim\Gamma(\frac{1}{2},1)$. $X_n$ and $Y_n$ ...
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What can we say about the sum of independent random variables with finite expectation?
Currently reading up on some measure-theoretic probability, and got stuck on the following problem from an earlier exam in a probability theory course.
Problem 3
Let $X_1, X_2, \ldots$ be a sequence ...
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Central limit theorem for uncorrelated and identically distributed random variables
I'm trying to determine whether a sum of identically distributed but only uncorrelated continous random variables could possibly converge to a normal distribution. First of all we define the i.i.d. ...
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Random walk on a k-dimensional grid
Consider a $k$-dimensional grid with integer points and let's begin our random walk at the origin $(0,0,\dots,0)$. Every step you have to move on each cartesian axis in the following way:
$x_1$ axis: ...
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Bootstrap Weak Convergence
Let $X_1, X_2, \dots, X_n$ be an iid sample from an unknown distribution finite mean $\mu$ and finite variance $\sigma^2$. Furthermore, let $R_1,R_2,\dots,R_n$. denote iid Rademacher random variables.
...
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What's the limiting distribution of the eigenvalue and eigenvectors of the sample covariance matrix?
The question is from the book 'Statistical Models and Methods for Financial Markets', Page 43.
Suppose $\boldsymbol{x_1,..., x_n}$ are $n$ independent observations from a multivariate population with ...
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$X_n$ is the number of heads in the sequence of tosses before the tails appear for the $n$-th time. Check if $\frac{X_n}{n}$ converges in probability.
We consider an infinite sequence of tosses of a symmetrical coin. For a natural number $n$, let $X_n$ denote the number of all the heads that appear in the sequence before the tails appear for the $n$-...
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Are the set of finitely supported probability measure in Euclidean space compact?
Suppose X is a compact subset of Euclidean Space. Let P(X) be the set of all Borel probability measures on X. Now let's consider the set of all Borel probability measures on X with finite support. My ...
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(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes
Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$,
\begin{equation}
\int_0^te^{-\lambda ...
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We have {$X_n$}, n$\ge$1 - a sequence of independent and equally distributed random variables with finite E|$X_1$|. We need to prove that...
The problem:
We have {$X_n$}, n$\ge$1 - a sequence of independent and equally distributed random variables with finite E|$X_1$|. We need to prove that $E|\frac{S_n}{n} - E$X_1$|\rightarrow 0$, where $...
user1184577
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Contradiction of key renewal equation applied to inter-arrival CDF
I'm looking for where I've made a mistake, since I'm reaching a contradiction after applying key renewal theorem and differentiating through the convolution.
Let $(X_t)$ be a renewal process with ...
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1
answer
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Prove the limit of the expectations of max over $X_i/n$ goes to $0$ for positive integer-valued r.v. with identical distributions but NOT independent
Let $X_1,X_2,...$ be positive integer-valued random variables with the same distribution but they can be dependent on each other. We want to prove that $\lim_{n\rightarrow \infty} E[\max_{i<n} X_i/...
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vote
1
answer
50
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Converge in probability
I'm trying to solve the following Probability problem:
Consider $ X(\omega) : [0,1] \to [0,1] $, where $f(x) = 2x \mathbb{I}_{[0,1/2]}(x) + 0.75 \mathbb{I}_{\{1\}}(x) $ and $X(\omega) = \omega = x$. ...
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2
answers
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Expectation of inverse of the sum of positive random variables
Suppose we have a sequence of independent identically distributed positive random variables $X_1,X_2,\cdots\stackrel{i.i.d.}{\sim}\xi$, and I am puzzled with the existence of expectation
$$\mathbb{E}\...
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0
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37
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Distribution of convergent sum of random variables [duplicate]
Let $\{X_n\}_{n=1}^\infty$ be an i.i.d. sequence of random variables that are distributed uniformly on $\{\pm 1\}$.
Define $$Y=\sum_{n=1}^\infty \frac{X_n}{n}$$ Can anything meaningful be said about ...
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1
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Convergence of a "simple" stochastic algorithm
In this question, we will consider the convergence of an algorithm as follows.
Let $f(x) = \frac{1}{2}x^2$ be our objective function. We define a so-called forcing function $\rho(\alpha) = \frac{1}{2}\...
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Probability of a volatile stock and its limiting behaviour
Each day, a very volatile stock rises 70% or drops 50% in price, with equal probabilities and different days independent. Suppose a hedge fund manager always invests a fraction $\alpha$ of her current ...
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