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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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Law of large numbers for non-identically distributed Bernoulli random variables

Let $(X_n)$ be a succession of independent r.v., such that $X_n$ ~ $Bern(p_n)$. I know then that $\lim_{n \to \infty}p_n=p$ and $p_n>p>0$ for each $n \in \mathbb{N}$. I have to prove that $\...
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17answers
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Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ...
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4answers
9k views

Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: ...
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2answers
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Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ ...
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3answers
775 views

Sums of independent Poisson random variables

The question is the following : $X_n$ are independent Poisson random variables, with expectations $\lambda_n$, such that $\lambda_n$ sum to infinity. Then, if $S_n = \sum_{i=1}^n X_i$, I have to ...
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2answers
314 views

Strong law of large numbers for function of random vector: can we apply it for a component only?

Consider i.i.d. random variables $\{X_1,..., X_n\}$ with well defined first moment i.i.d. random variables $\{Y_1,..., Y_n\}$ with well defined first moment By the strong law of large numbers: $$ \...
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0answers
398 views

The law of large numbers with dependent random variable

Consider a sequence of i.i.d. random variables $\left\{X_i\right\}_i$, and let $Y$ be another random variable. Can we say something regard the convergence of the following series $$ \frac{1}{n}\sum_{...
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4answers
925 views

What happens if I toss a coin with decreasing probability to get a head?

Yesterday night, while I was trying to sleep, I found myself stuck with a simple statistics problem. Let's imagine we have a "magical coin", which is completely identical to a normal coin but for a ...
3
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1answer
437 views

If $(X_n)$ is i.i.d. and $ \frac1n\sum\limits_{k=1}^{n} {X_k}\to Y$ almost surely then $X_1$ is integrable (converse of SLLN)

Let $(\Omega,\mathcal F,P)$ be a finite measure space. Let $X_n:\Omega \rightarrow \mathbb R$ be a sequence of iid r.v's I need to prove that if: $ n^{-1}\sum _{k=1}^{n} {X_k} $ converges almost ...
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1answer
304 views

Strong Law of Numbers for $S_{n}$ Bounded Casella Berger 5.38

So this is from Casella Berger 5.38 b) the question states `` Let $X_{1},...,X_{n}$ be iid with mgf $M_{X}(t)$. Let $S_{n} = \sum X_{i}$ and $\bar{X_{n}}= \frac{S_{n}}{n}$. use the fact that $M_{X}...
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1answer
91 views

Strong law of large numbers for a scaled sequence of normally distributed random variables

Let $f\in C^3(\mathbb R)$ be positive $g:=\ln f$ $d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^...
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1answer
720 views

Why weak law of large number still alive?

I know the difference between WLLN and SLLN in terms of a convergence type. Then, as revealed in any statistical textbook saying sufficient conditions to two theorems are the same, I think that we do ...
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0answers
121 views

Distribution of sums of inverses of random variables uniformly distributed on [0,1]

If I have $N$ random variables (denoted below as $X_i$) with uniform distribution on the $x$-axis $X_i = \rm{rand}[0,1]$ then the sum $$ S_N = \frac{1}{N}\sum_i^N\frac{1}{2X_i-1} $$ seems to be a ...
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1answer
941 views

Weak/strong law of large numbers for dependent variables with bounded covariance

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of $L^2$ random variables with expected value $m$ for all $n$. Let $S_n=\sum_{i=1}^n X_i$ and $|\mathrm{Cov}(X_i,X_j)|\leq\epsilon_{|i-j|}$ for finite, non-...
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2answers
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Why aren't the strong LLNs and CLT contradicting each other?

Given $n$ i.i.d. random variables $\{X_1, X_2, \dots , X_n\}$, each with mean $M$ and variance $V$, both strong and week LLNs seem to say that the average of the $n$ random variables, $S_n = \frac{X_1 ...
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1answer
275 views

Kolmogorov's sufficient and necessary condition for SLLN - What about pairwise uncorrelated RV?

Kolmogorov proved, that, as one considers independent (not necessary equally distributed) Random Variables: $\{X_n\}_{n\ge0}\subseteq \mathcal L^2$ With $\mathrm{Var} (X_n)=\sigma^2_n$ and without ...
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1answer
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Is the distribution of a product of M iid uniform random variables really Log Normal?

Conventional wisdom says yes or mostly. But consider the following simple derivation: Let $y = \prod_{i=1}^M x_i$ where $x_i\sim U(0,1)$. Then from independence, $E[y] = 2^{-M}$. Now, if we let $...
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1answer
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a.s. convergence of sum of normal random variables

From Resnick's A Probability Path, Exercise 7.7.14: Suppose $\{X_n, n \ge 1\}$ are independent, normally distributed with $E(X_n) = \mu_n$ and Var$(X_n)=\sigma^2_n$. Show that $\sum_n X_n$ ...
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2answers
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using Chebyshev's inequality

Suppose that you take a random sample of size n from a discrete distribution with mean $\mu$ and variance $x^2$. Using Chebyshev's inequality, determine how large n needs to be to ensure that the ...
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0answers
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A Problem on the Limit of a Sequence

While I investigate the property of positive random variables, I encountered the following question, not easy to solve. Let $\left\{ a_n \right\}_{n=1}^{\infty}$ be a sequence that satisfies $a_n &...
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1answer
368 views

Proving almost sure convergence

Assume the sequence of random variables $X_1, X_2, \cdots$ are IID with finite mean and finite variance. Define a random variable: \begin{align} Y_n = \frac{X_n}{n} \end{align} Show that $Y_n \to 0$ ...
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1answer
586 views

Triangular arrays and almost sure convergence of row averages

Suppose we have the triangular array $\{\{X_{in},i=1,\ldots,n\},n=1,2,\ldots\}$: $$\begin{array}{ccccc} X_{11}&&&&\\ X_{12}&X_{22}&&&\\ X_{13}&X_{23}&X_{33}&...
5
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1answer
107 views

Law of Large Numbers - utility/difficulty of various versions.

This may or may not be an answer to Is there an easy proof that the set of $x \in [0,1]$ whose limit of proportion of 1's in binary expansion of $x$ does not exist has measure zero?, depending on ...
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5answers
1k views

Law of Large Numbers, a confusion

According to Law of Large Numbers, if I throw a coin 1000 times approximately 500 will be head and 500 tail. Suppose that I throw the coin 700 times and I got 700 heads. Can I say that in the next 300 ...
2
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1answer
24 views

$X=\sum_{i=1}^{N}X_i$,estimator for $N$(continuation)

Let $X_i$, $i\geq 1$, be independent and identically distributed random variables having the uniform distribution over $(0,1)$. Let $X$ be defined as $X=\sum_{i=1}^{N}X_i$, where $N$ is an unknown ...
2
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1answer
61 views

If $(W_i)_{i\in\mathbb N}$ obeys the strong law of large numbers, what can we say about $\liminf_{d\to\infty}\frac1{d^{2\alpha}}\sum_{i=1}^dW_i$?

Let $d\in\mathbb N$ and $W_1,\ldots,W_d$ be mutually independent, identically distributed and square-integrable real-valued random variables on a probability space $(\Omega,\mathcal A,\operatorname P)$...
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1answer
51 views

$X=\sum_{i=1}^{N}X_i$,estimator for $N$

Let $X_i$, $i\geq 1$, be independent and identically distributed random variables having the uniform distribution over $(0,1)$. Let $X$ be defined as $X=\sum_{i=1}^{N}X_i$, where $N$ is an unknown ...
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2answers
103 views

Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = Y_{n-1}(1+...
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1answer
115 views

limit of average of independent, but not identically distributed r.v.

Let $\{X_i\}$ be a collection of independent r.v., but with distribution dependent on index $i$, such that $P(X_i=2^i)=2^{-i}$ and $P(X_i=0)=1-2^{-i}$ for $i \in \mathbb{N}$. What can I say about $\...
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1answer
138 views

Strong law of large numbers with $\sum_{n=0}^\infty \frac{Var[S_n]}{n^2}<\infty$

Given independent real random variables $X_1,X_2,... \in L^2$ with $$\sum_{n=0}^\infty \frac{Var[S_n]}{n^2}<\infty$$ (here $S_n := X_1+...+X_n$). How do you show that $(X_n)_{n \in \Bbb N}$ ...
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0answers
535 views

How to use Chebyshev's inequality or the law of large numbers to a probability question?

Let $x$ be a random bit string that takes values $\{1,0\}^n$. Let $r$ be the value of the most significant bit (MSB) of $x$ (and $r$ is a r.v. $1$ or $0$ that are equally likely). Let $g$ be our guess ...
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1answer
157 views

How is this not a counter-example to the law of large numbers?

Let $\Omega = \{0,1\}$ and $X: \Omega \rightarrow \{0,1\}$ be a random variable s.t. $X = id$ with $E[X] = 0.5$ (i.e., $P(0) = 0.5 = P(1)$). Let $X_1$, $X_2$, $\ldots$ be a sequence of i.i.d. random ...
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1answer
2k views

Showing convergence in probability of sample variance to population variance

Problem 6.7.2 of Resnick's A Probability Path says: Let $\{X_{n}\}$ be iid, with $EX_{n}=\mu$, $\mathrm{Var}(X_{n})=\sigma^2$. Set $\overline{X}=\sum_{i=1}^{n} X_{i}/n$. Show: $$ \frac{1}{n}\sum_{i=1}...
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1answer
192 views

Law of large numbers for nonnegative random variables [closed]

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. ...
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1answer
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what percentage of the integers in the range $l,l+1,\dots k$ contain a certain digit?

what percentage of the integers in the range $l,l+1,\dots k$ contain a certain digit? For example, say l is 0, k is 9, and my ...
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1answer
51 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\...
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2answers
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Questions on Inverting Laplace transforms and Probability

From Williams' Probability w/ Martingales: Are we allowed to switch derivative and integral as follows: $$\frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x} f(x) = \int_{0}^{\infty} \...