Questions tagged [law-of-large-numbers]
For questions about the law of large numbers, a classical limit theorem in probability about the asymptotic behavior (in almost sure or in probability) of the average of random variables. To be used with the tags (tag:probability-theory) and (tag: limit-theorems).
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Confusion about LLN for Empirical Processes/Measures
I have seen the following LLN result on empirical measures:
(Statement 1) Let $X, X_1, X_2, \cdots, X_n$ be i.i.d. random variables taking values in $[0, 1]$. Then
$$
\mathbb{E}\sup_{f \in \mathcal{F}...
- 551
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"Elementary" Proof of Strong Law of Large Numbers
I came across a paper that a seemingly "elementary" proof of the strong law of large numbers, although I am having a lot of trouble understanding it since the author leaves a lot of ...
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If $\sup_{f\in F} \Big|||f||_n-||f|| \Big|\overset{p}{\to} 0$ then $\sup_{f\in F, ||f||\neq 0} \Big| ||f||_n/||f||-1 \Big| \overset{p}{\to} 0$.
SETUP:
Let $\{X\}_{t\in\mathbb{N}}$ be a strictly stationary stochastic process on the probability space $(\Omega, \mathcal{F}, P)$ where for each $t\in\mathbb{N}$ the random elements are such that $...
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How can an estimator be consistent and asymptotically normal at the same time?
I can't work out why the asymptotic distribution of an estimator matters if it is consistent?
My understanding is:
An estimator, $\hat{\theta}$, is consistent if it converges in probability to the ...
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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 ...
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Law of large number with subset of the variables
Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}\subset\mathbb{R}$ be decreasing ...
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What is the Asymptotic Order of the Sum of Random Variables with non-finite Moments?
Suppose $X_i$ is independently and identically distributed over $i$, and $E(|X_i|^d)$ with $d>0$ is not finite or undefined:
I am wondering whether $\sum_{i=1}^N |X_i|^d$ is of any asymptotic ...
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For a random binary sequence of length $2N$ with $n$ 1's, what is the probability that the first $N$ digits have a 'similar' proportion of 1's?
More precisely, if we know the proportion of 1's in a uniformly randomly chosen binary sequence of length $2N$ is $p$, then given an $\epsilon>0$ , what is the probability that the proportion of 1'...
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Understanding a step in this proof of the $L^4$-SLLN
Let $(X_n)_{n \ge 1}$ be independent with $E(X_k) = \mu$ and $E((X_k)^4) \le C$ for some $C>0$. Prove that if $S_n = X_1 + ... + X_n$ then $\frac{S_n}{n} \to \mu$ almost surely as $n \to \infty$.
...
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Difference of hypotheses between weak and strong law of large numbers
I have read several questions on the difference between weak and strong laws in terms of convergence in probability vs convergence a.s. and I think I grasp them.
However, I have uncertainty on the ...
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A lemma involving Chebyshev's inequality to prove the weak law of large numbers
Reading Roch's Modern Discrete Probability one finds the following exercise (2.6):
(Sums of uncorrelated variables). Centered random variables $X_1,\dots X_n$ are uncorrelated if forr all $r$ and for ...
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Monotonicity of the probability of a sum of independent random variables being below a threshold
Suppose I have a sequence of i.i.d. random variables $X_1, X_2, \dots$ with positive mean $E[X_1] = \mu$. For $A>0$, is the function
$$f(n) = \Pr( X_1 + X_2 + \dots + X_n \leq A)
$$
monotonically ...
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Central Limit Theorem for Difference-in-Means Estimator
I am studying Lecture 1 of Stefan Wager's Causal Inference notes and come across a central limit theorem for the difference-in-means estimator, which I am unable to prove.
The mathematical abstraction ...
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Consistency of Biased Estimators
In Statistical Inference, we were taught this theorem,
Consider an estimator $T_n$ of population parameter $\theta$, using $n$ samples. $T_n$ is a Consistent Estimator of $\theta$ if $$E[T_n] \to \...
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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 ...
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Application of ergodic theorem to transformed Gaussian process
Assume that $W$ is a zero-mean Gaussian process on $[0,\infty)$ with covariance function satisfying $c(s,t):=E(W(s)W(T))=C(|t-s|)$ for a function $C:[0,\infty) \to \mathbb{R}$.
Can I exploit somehow ...
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Law of large numbers for non-independent and non-identically distributed samples
Let $X \sim p_X$ be a real-valued random variable with $\mathbb{E}[X] = \mu > c$ where $c \in \mathbb{R}.$
Assume you sample from $p_X$ and only accept samples such that the current sample mean is ...
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Weak law of large numbers for distributed Bernoulli random variables in a particular case. [closed]
Let $X_1, X_2, ...$ be a sequence of independent and identically distributed random variables, such that $X_i\sim Bern(p)$. Now, let $Y_{1,n}, Y_{2,n}, ...$ be random variables, such that $Y_{i,n}\sim ...
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Empirical Distribution Convergence, Ordering Of Samples
I am trying to formally justisty a "rearrangement" algorithm, which rearranges the samples of two random variables to reflect a certain joint distribution.
Suppose that we have two pairs $(...
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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 ...
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How to understand the proof of the division theorem in the four arithmetic operation theorems of probability convergence for random variables?
\begin{equation}
Y_n \stackrel{P}{\longrightarrow} b \text {}
\end{equation}
and:
\begin{aligned}
& P\left(\left|\frac{1}{Y_n}-\frac{1}{b}\right| \geqslant \varepsilon\right)=P\left(\left|\frac{...
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Suppose $(X_{n})_{n}$ is series of i.i.d. random variables and $E(X_{1})=2$. Compute $\lim_{n\to\infty} (n \ln (\varphi_{X_{1}}(\frac{2}{n})))$
Let's define $S_{n}=\sum_{k=1}^{n} X_{k}$. According to strong law of large numbers we have $\frac{S_{n}}{n} \rightarrow E(X_{1})=2$ a.s. It follows from that $\frac{S_{n}}{n} \rightarrow 2$ in ...
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Inifinite numbers of i.i.d Markov chain and law of large number.
Let $(X(t))_{t \geq 0}$ be a continuous time-homogeneous Markov chain with values in the state space $S$ that we assume to be finite (countable should also work). We denote $P = (p_{ij})_{ij}$ the ...
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Quantitative law of large numbers for non-identically distributed random variables
I don't know much about probability so this may be an easy consequence of some well known theorem.
Suppose we have a sequence of independent, $\mathbb{C}$-valued random variables $(\xi_n)_{n\in\...
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Law of Large Number for Stochastic Processes
Consider the following Stochastic Process:
$$B(t)∼N(μt,σt)$$
Here is a simulation for multiple possible trajectories of this Stochastic Process (R Code):
...
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How do I prove that the strong law of large numbers is equivalent to $P(∩_{n=1}^∞ ∪_{{k}\geq{n}}{|{\frac{1}{k}}S_k-p|\geq{ε}})=0$
I want to solve the following task:
Show that the strong law of large numbers statement is equivalent to the following statement:
For all ε>0: $$P(∩_{n=1}^∞∪_{{k}\geq{n}}{|{\frac{1}{k}}S_k-p|\geq{ε}...
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Convergence of a capital describing random variable
In my previous question Infinite game of an unfair coin toss, I showed that in a game between Person A and B, where:
Person A has an unfair coin with probability $p \in (\frac{1}{3},\frac{1}{2})$ of ...
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Weak Law of Large Numbers (Non-convergence in probability)
Given $X_1, X_2,..., X_n$ independent random variables with $P(X_n=4^n)=P(X_n=-4^n)=1/2$. Let $S_n = X_1 + X_2 +... + X_n$. Determine for which $\epsilon>0$, if any, that the $P(|S_n|/n>\epsilon)...
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Prove or Disprove Almost Sure Convergence
Let $𝑋_1,𝑋_2, … , 𝑋_𝑛$ be a sequence of random variables, with $𝐸(𝑋_𝑛) = 7$ and $𝑉𝑎𝑟(𝑋_𝑛) =
1/sqrt(𝑛^𝑎)$ for each 𝑛. Prove or disprove that ${𝑋_𝑛}$ must converge to 7 with probability ...
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Application of Weak Law of Large Number
Let $𝑋_1,𝑋_2, … , 𝑋_𝑛$ be a sequence of independent random variables with $𝑃(𝑋_𝑛 = 4^𝑛) =
𝑃(𝑋_𝑛 = −4^𝑛) = \frac12$. Let $𝑆_𝑛 = 𝑋_1 + 𝑋_2 + ⋯ + 𝑋_𝑛$. Determine for which $𝜀 > 0$, ...
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Converse of the weak law of large numbers [closed]
I solved a nice exercise which consists of proving a converse for the law of large numbers which is, if $(X_i)_{i \in\mathbb{N}^*}$ are iid random valued such that $$ \frac{X_1+\ldots+X_n}{n} \to Y$$ ...
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Asymptotic convergence of a count-distinct estimator
Consider the following variant of the count-distinct problem.
Problem Setting. Let $B = \{ b_1, \dots, b_n \}$ be a finite set of $n$ balls, let $C$ be a set of colours, let $F : B \to C$ be a ...
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Strong law of large numbers for $\mathrm{Bin}(n, p_n)$ variables
Massive edit to simplify the central question
Suppose $X_n\sim \mathrm{Bin}(n, p_n)$ be a collection of independent random variables such that $np_n\to \infty$. Can we say that $Y_n:=X_n/np_n\to 1$ ...
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Sufficient Conditions for Strong Law of Large Numbers
In this paper, I found that if $\lbrace X_i \rbrace$ is an independenly indentically distributed (i.i.d.) sequence and $E(|X_i|)<\infty$, then $\bar{X}_{N}=\sum_{i=1}^N\frac{X_i}{N}$ converge to $E(...
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sum of two point distribution converge to infinity almost surely
Question:
$\{X_k\}_{k=1}^{\infty}$ i.i.d.,$P(X_1=-1)=P(X_1=1)=\frac12$.
$S_n=\sum\limits_{k=1}^n X_k$.
Prove that:
1.$S_n$ converge to infinity almost surely,i.e. $P(\lim_{n\to+\infty}S_n=\infty)=1$;
...
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Weak vs strong law of large numbers
Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of real i.i.d. random variables with mean $\mu$. Let $S_n$ be the sum of the first $n$ elements of this seqeuence,
$$S_n = \frac1n\sum_{i=1}^n X_i.$$
...
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Convergence in Probability (Law of Large Numbers)
Suppose $X_1,X_2,…,X_n$ are iid Poisson random variables, each with mean $\theta$. How to prove that $Y_n=\exp[−\frac{1}{n}(X_1+X_2+⋯+X_n)]$ converges in probability to $P(X=0)=\exp(−\theta)$ ?
Hint: ...
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Convergence in Probability (Weak Law of Large Number)
Suppose $𝑌_1, 𝑌_2, … , 𝑌_𝑛$ are independent and identically
distributed Poisson random variables, each with mean $𝜆$. Prove that $𝑋𝑛 =exp[−(1/𝑛)(𝑌_1 + 𝑌_2 + ⋯ + 𝑌_𝑛)]$ converges in ...
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Epsilon-Delta analysis for the Law of Large Numbers?
Law of Large Numbers: For a sequence of independent and identically distributed random variables $X_1, X_2, X_3, ..., X_n$, each with an expected value $E[X_i] = \mu$ and sample estimator $\overline{...
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Computing the expectation of random variable by marginalizing over the condtional expectation and prove with LLN
I am trying to estimate the expectation of $X$ by marginalize over $Y$ and the conditional expectation $E(X|Y)$ and prove that it converges to the true expectation of $X$ as the sample size approaches ...
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$\{X_k\}$ are iid (the standard normal distribution), what is the distribution of $\lim \tau_n=\frac{X_{n+1}}{\sqrt{\sum_{k=1}^n X_k^2/n}}$?
$\{X_k\}_{k=1}^{\infty}$ are independent and identically distributed
(the standard normal distribution),
$$Y_n^2=\sum_{k=1}^n X_k^2, \ \ \
\tau_n=\frac{X_{n+1}}{\sqrt{Y_n^2/n}}.$$
What is the ...
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Confusion on one step in proof in Achim Klenke's Probability Theory
I've come to the following theorem and proof in Klenke's Probability Theory book, and I've hit a roadblock on the very last step of the proof. We know $\limsup_{n\to\infty}|k_n^{-1}S_{k_n}-\mathbf{E}[...
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Proof of Marcinkiewicz-Zygmund strong law of large numbers
Marcinkiewicz-Zygmund strong law of large numbers:
Let $X_1,X_2,···$ be i.i.d. with $E|X_1|^p< \infty$ for some $0< p <2$. Then
$$
\begin{cases}
\frac{S_n - nEX}{n^{1/p}} \to 0 \text{ a.s.} &...
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Seeking help with the application of Law of Large Numbers and Central Limit Theorem to calculate Investor Risk
I'm a newbie to the forum with zero financial or statistical skills - first time post...seeking some assistance and a solution..thanks in advance!
I am trying to create a investor calculator or at ...
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$\text{Var}(X_n) < \infty$ for all $n$, $\text{Var}(S_n) = o(n^2)$ when $n \to \infty.$ Prove $\frac{S_n}{n} \to E(S_n)$ in probability.
Random variables $X_n$ are independent, but not identically
distributed. $\text{Var}(X_n) < \infty$ for all $n$, $\text{Var}(S_n) = o(n^2)$ when
$n \to \infty.$ Prove $\frac{S_n - E(S_n)}{n} \to
0$ ...
- 1,207
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Step in the proof of the strong law of large numbers.
Let $(X_i)_{i \geq 1}$ i.i.d. and $\mathbb{E}(|X_1|)< \infty$, w.l.o.g. $\mathbb{E}(X_1) = 0$. This in turn implies:
$$
\sum_{n=1}^{\infty} \mathbb{P}(|X_1| \geq n) \leq \int_0^{\infty} \mathbb{P}(|...
- 1,297
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Will the expanding cube look gray?
Consider the tiling of $\mathbb{R}^n$ by unit cubes centered at integer lattice points, i.e.
$$ \mathbb{R}^n = \bigcup_{a \in \mathbb{Z}^n} Q\left(a, \frac{1}{2}\right). $$
Color each unit cube black ...
- 1,007
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Strong Law of Large Numbers for increasing index sets
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of integrable i.i.d random variables. Let $I_n \subset \mathbb{N}$ satisfy $\lim_{n \to \infty} |I_n| = \infty$, where $|I_n|$ is the cardinality of $I_n$. ...
- 1,007
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Law of large numbers for part sum of terms in random integer composition
Let $C=C_{n,m}=(c_1,\ldots,c_n)$ be a random weak $n$-part integer composition of $m$ drawn uniformly from all $\binom{m+n-1}m$ such compositions. Suppose $m\to\infty$ as $n\to\infty$ (e.g. $m\sim n^\...
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The Money Left in an Infinite Gambling Game
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Suppose that $\Omega$ is defined as below
\begin{equation*}
\Omega = \Big\{ \omega = (\omega_1, \omega_2, \dots) \big|\, \omega_i \in \{...
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