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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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 $E[X_1] = \mu$. Is the function $$f(n) = \Pr( X_1 + X_2 + \dots + X_n \leq A) $$ monotonically non-increasing ...
ilanshom's user avatar
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18 views

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 ...
Marlovo's user avatar
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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 \...
Harry's user avatar
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2 answers
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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 ...
Learning Math's user avatar
2 votes
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35 views

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 ...
Jack London's user avatar
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A sequence of PDFs converges to a random variable [closed]

Let ${f_b(x)}:[0,1] \to [0,1]$ be a set of PDFs, where $b\in[0,1]$ is a parameter. Assume that $f_b(x)$ is continuous in $b$ for every $x$. Let $X_i$ be a sequence of random variables such that $X_i\...
yjjjant's user avatar
2 votes
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82 views

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 ...
tobayes's user avatar
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1 answer
29 views

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 ...
Helder Alves Arruda's user avatar
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24 views

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 $(...
Nicola Zaugg's user avatar
3 votes
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74 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
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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{...
SUNi's user avatar
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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 ...
bnagy01's user avatar
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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 ...
Velobos's user avatar
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2 votes
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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\...
Saúl RM's user avatar
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1 vote
2 answers
241 views

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): ...
Uk rain troll's user avatar
1 vote
0 answers
41 views

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}...
MathStudentRUB's user avatar
2 votes
1 answer
85 views

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 ...
tychonovs-scholar's user avatar
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0 answers
26 views

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)...
tt99999's user avatar
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1 answer
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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 ...
john22445's user avatar
1 vote
0 answers
72 views

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 $𝜀 > 0$, ...
john22445's user avatar
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1 answer
76 views

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 $$ \frac{X_1+\ldots+X_n}{n} \to Y$$ ...
Rafaël's user avatar
  • 181
3 votes
2 answers
153 views

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 ...
Boris's user avatar
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4 votes
2 answers
90 views

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 $Y_n:=X_n/np_n\to 1$ ...
Landon Carter's user avatar
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0 answers
80 views

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(...
Jack's user avatar
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0 answers
67 views

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. $P(\lim_{n\to+\infty}S_n=\infty)=1$; ...
shdvt's user avatar
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2 votes
1 answer
70 views

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, $$S_n = \frac1n\sum_{i=1}^n X_i.$$ ...
CBBAM's user avatar
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1 vote
1 answer
123 views

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: ...
john22445's user avatar
0 votes
1 answer
160 views

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 ...
john22445's user avatar
4 votes
1 answer
104 views

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{...
Uk rain troll's user avatar
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0 answers
32 views

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 ...
Coodyyy's user avatar
1 vote
1 answer
37 views

$\{X_k\}$ 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 ...
fragileradius's user avatar
0 votes
1 answer
37 views

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}[...
modz's user avatar
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2 votes
1 answer
79 views

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.} &...
Fireond's user avatar
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0 votes
0 answers
32 views

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 ...
Darren's user avatar
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1 vote
0 answers
43 views

$\text{Var}(X_n) < \infty$ 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 $\frac{S_n - E(S_n)}{n} \to 0$ ...
fragileradius's user avatar
0 votes
1 answer
67 views

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}(|...
Henry T.'s user avatar
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2 votes
0 answers
70 views

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 ...
Cauchy's Sequence's user avatar
5 votes
1 answer
109 views

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 $I_n$. ...
Cauchy's Sequence's user avatar
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0 answers
42 views

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^\...
David Bevan's user avatar
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1 vote
1 answer
113 views

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 \{...
Hosein Rahnama's user avatar
0 votes
1 answer
33 views

Given $X_n$ iid random variables with $P(X_n = n^2) = 1/n^2, P(X_n = -1) = 1-1/n^2$ show that $\bar{Y_n} \xrightarrow{P} 0$ where $Y_n = X_n/n^2$

Given $X_n$ iid random variables with $P(X_n = n^2) = \frac{1}{n^2}, P(X_n = -1) = 1-\frac{1}{n^2}$. Let $Y_n = \frac{X_n}{n^2}$, I want to show that $\bar{Y_n} \xrightarrow{P}0$, where $\bar{Y_n} = \...
Rahid Fayad's user avatar
0 votes
0 answers
63 views

Why are these definitions of the Law of large numbers the same?

I have learnt the following definition of the law of large numbers: Theorem 17.4 (Law of Large Numbers). Let $X_1, X_2, \ldots$, be a sequence of independent and identically distributed random ...
Princess Mia's user avatar
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0 votes
1 answer
60 views

Who needs the law of large numbers when you have the central limit theorem

The central limit theorem (CLT) and law of large numbers (LLN) are both statements about the sample mean. If the sample mean of $n$ samples is $\bar{X}_n$, the CLT says that the distribution of: $$\...
Rohit Pandey's user avatar
  • 6,731
1 vote
1 answer
70 views

limit and integral / strong law of large

I am currently calculating $$\lim_{n \to \infty} \int_{[0,1]^n} f\left(\frac{x_1+...+x_n}{n}\right)\mathrm{d}x_1...\mathrm{d}x_n$$ with $f$ a continuous function between $[0,1]$. We deduce that $$\...
Edmond's user avatar
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0 votes
0 answers
91 views

Measure of the set of numbers in $[0,1]$ expressed with different quantities of zeroes and ones in binary

I read a Math Overflow answer recently that claimed the set of numbers in $[0,1]$ whose binary expansion contained a different number of 0 and 1 digits (i.e. doesn't include numbers like $0.\overline{...
Andrey Shlykov's user avatar
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0 answers
54 views

The asymptotic distribution of the criterion function in M-estimation

Suppose that we are interested in a parameter (or functional) $\theta$ attached to the distribution of observations $\left\{ X_{1},\ldots,X_{n}\right\} $. A popular method for finding an estimator $\...
Hagan Ross's user avatar
3 votes
1 answer
185 views

Polya's urn, should I use martingales or LLN

I am trying to prove the following question, but I am finding it a bit tricky to determine the distribution of $X_i$ (the number of red balls drawn in the $i$-th round) and thus I do not know which ...
idlatva's user avatar
  • 153
2 votes
1 answer
27 views

Help with starting a proof regarding empirical distribution function of a uniform distribution

Suppose $U_1,...,U_n$ is a simple sampling from a uniform distribution $U(0,1)$ and $G_n(u)$ is an empirical distribution function. Prove that $$ \begin{gathered} n \int_0^1\left(G_n(s)-s\right)^2 d s=...
1ncend1ary's user avatar
0 votes
0 answers
37 views

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 \...
P.S. Dester's user avatar
-1 votes
1 answer
130 views

The weak law of large numbers and random graphs

Recently, I am learning about the weak law of large numbers (WLLN) and I find some of its interesting applications on random graphs. The question is as follows. Given $n$ vertices labelled by $\{ 1, 2,...
MATHQI's user avatar
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