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).
619
questions
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1answer
13 views
An upper bound to $\mathbb{P}[|\sum_{i=1}^{n}(X_i - \mu)| > \epsilon n] $
Let $\epsilon > 0, \, (X_i)_{i\geq1}$ be a sequence of identically distributed (we denote $EX_i$ by $\mu$) and pairwise (not mutually!) independent random variables
prove that there exists some ...
0
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0answers
16 views
Prove markov condition of WLLN holds when both WLLN and CLT hold for a series of independent random variables
The original question is as follows:
$\{X_k\}$ are independent random variables, and the Central Limit Theorem holds, which means $\dfrac{\sum_{i=1}^{n} (X_i-EX_i)}{\sqrt{\sum_{i=1}^n Var(X_i)}} \...
1
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0answers
31 views
Prove SLLN doesn't hold for a given series of random variables
The random variable series $\{X_k\}$ is as follows:
$P(X_k=\pm k)=\frac{1}{2}k^{-\frac{1}{2}}, P(X_k=0)=1-k^{-\frac{1}{2}}$. The variables are independent. I need to prove that the Strong Law of Large ...
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0answers
14 views
Find conditions on the constants $L_k$ which will ensure that the law of large numbers and/or the central limit theorem holds for $\{X_k\}$.
(Feller Vol 1, Q.4, P.261) Let the $X_k$ be mutually independent random variables such that $X_k$ assumes the $2k+1$ values $0, \pm L_k, \pm2L_k, ... , \pm kL_k$, each with probability $1/(2k+1)$. ...
1
vote
1answer
14 views
Probability that most likely outcome will appear more often in finite sequence
Let $(X_n)_{n\geq 1}$ be an i.i.d. sequence of variables with unequal Bernoulli distribution : $P(X=0)=p, P(X=1)=1-p$ with $p\gt \frac{1}{2}$ (so getting a $0$ is more likely than getting a $1$).
Let ...
0
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1answer
22 views
Weak Law for Dependent Sequence with bounded variance
I was wondering if the Weak Law of Large Numbers also holds true for dependent identically distributed random variables with constant mean and bounded variance.
Let $X_1,X_2\dots X_n$ be a sequence ...
0
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0answers
8 views
Rate of convergence for the law of large numbers if Multivariate Random variables
I am looking for a reference on rates of convergence for the law of large numbers on random variables on $R^d$ for $d>1$.
I haven't been able to find (or find very little) about LLN of ...
3
votes
1answer
42 views
Using the Strong Law of Large Numbers to find a constant, c.
Let $X_1, X_2...$ be independent and identically distributed with mean 4 and variance 20. Set $S_n = X_1 + X_2 + ... + X_n$ and $V_n = X_1^2 + X_2^2 + ā¦ + X_n^2$. Use the Strong Law of Large Numbers ...
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0answers
46 views
Showing $\frac{S_n}{n^{1/p+\varepsilon}}$ converges to $0$ a.e.
Let $\{X_n\}$ be independent and identically distributed random variables such that $E(|X_1|^p) < \infty $ for some $p$ satisfying $0 < p < 2$; in case $p > 1$, we assume that $E(X_1) = 0$,...
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2answers
35 views
How do Chebyshev inequality and Weak Law of Large Number translate into this?
The Chebyshev inequality states that if $X$ is a random variable with mean $\mu$ and variance $\sigma^2$, then:
$$\mathbf P\bigl(|X -\mu |\geq c\bigr)\leq\frac{\sigma^2}{c^2},~~~~\textrm{for all}~c &...
0
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0answers
32 views
What does the law of large numbers say about this sample average?
Let $X_1, X_2, \dots X_n$ be independent uniform random variables with range $X_i \in [ā1,2]$. What does the law of large numbers tell us about the value of
$$\frac{1}{n} \sum_{i} \lvert X_i \rvert
$$...
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0answers
26 views
Is there a way to calculate log of some number, let's say 2, if the base of logarithm is a very large number, say Graham's number?
Graham's number definition from: https://en.wikipedia.org/wiki/Graham%27s_number
Graham's number=G64
Of course the equation could be transformed in $\operatorname{G64}^x=2$, but I don't know how to ...
2
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0answers
21 views
How to estimate the mean and variance of the minimum of a random set?
Given $n$ numbers $x_i,i\in\{1,2,...,n\}$, define the set of all possible summation of $x_i$ with different signs:
$$A = \Big\{\Big|\sum_i x_i s_i\Big| : s_i = \pm 1\Big\}$$
and define the minimum of $...
3
votes
1answer
55 views
Strong law of large numbers for triangular arrays
Consider a triangular array $X_{n,1},\ldots,X_{n,n}$ of rowwise i.i.d. real random variables with $ \sup_{n \in \mathbb{N}} \mathbb{E}\vert X_{n,1} \vert < \infty$ and $ \lim_{n \rightarrow \infty}...
3
votes
1answer
82 views
Sum Infinite Random Variables
Let's say we generate $n$ samples independently from two independent distributions $X$ and $Y$. We know that the following is true from Jensen's Inequality:
$$\
E\left[\min\left(\sum_{i=1}^{n}X_i, \...
2
votes
0answers
26 views
The law of large numbers
I'm currently writing a dissertation on the law of large numbers. The 4th chapter is on the real-life applications of the laws themselves. I have found applications for the laws in general but would ...
4
votes
1answer
165 views
Law of large numbers holding uniformly with respect to a distribution
Let $X$ and $\varepsilon$ be independent random vectors, $\mathcal{X} = \text{supp}(X)$, and $Y = f(X) + \varepsilon$ for some function $f$.
For any $x \in \mathcal{X}$, let $y^i = y^i(\omega)$, $i \...
2
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0answers
39 views
Determine or find an upper bound for the limsup of a function with random variables
Determine or find an upper bound for
$$\limsup_{n \to \infty} \left|\frac{\frac{2}{n} (l(\beta) - l(\hat{\beta})) - (\beta - \hat{\beta})^T \nabla^2 l(\hat{\beta})(\beta - \hat{\beta})}{(\beta - \hat{...
1
vote
1answer
29 views
Strong Law of Large Numbers and Uniformly Distributed Random Numbers
Let $X_j$ be i.i.d. $\mathcal{U}[0,1]$ random variables. Prove that $\lim\limits_{n \to\infty} \frac{n}{X_1^{-1}+\dots+X_n^{-1}}$ exists almost surely and find the limit.
I think I need to use the ...
2
votes
1answer
23 views
Strong Law of Large numbers for continuous martingales
In the discrete case, we have the following theorem: Let $M_n$ be a squared-integrable martingale with quadratic variation $\langle M\rangle$. Then, $\displaystyle\frac{M_n}{\langle M\rangle_n}\...
2
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0answers
29 views
Possible application of Laws of Large Numbers and Central Limit Theorem to SVD of dual covariance matrices
Let $X:=[x_1\dots x_n] \in \mathbb{R}^{d\times n}, x_i \in \mathbb{R}^{d\times 1}$ be a data matrix where $x_i \in \mathbb{R}^{d\times 1}$ are iid random vectors with mean $\mu$ and covariance $\...
2
votes
0answers
28 views
Using conditional expectation in SLLN
In my notes we have $\mathbb{E}[\hat{Y}_n\mid\mathcal{F}_t]=X_t$ where $\hat{Y}_n=(Y^1_T+\cdots+Y^n_T)/n$ and $\mathbb{E}[Y^i_T \mid \mathcal{F}_t]=X_t$. We say that $\hat{Y}_n$ is unbiased, but I ...
2
votes
2answers
76 views
Convergence of empirical means in other norms
Say that we have a sequence $x_1,x_2,\dots$ of i.i.d. random vectors in $\mathbb{R}^n$ with mean $0$ and variance $\sigma^2$, meaning
$$\mathbb{E}[\|x_i\|^2_2] = \sigma^2$$
for all $i$. Then it's a ...
2
votes
0answers
46 views
Using a modification of the weak law of large numbers to conclude the fair price of the St. Petersburg game
I am reading Durrett's "Probability: Theory and Examples" section on the St. Petersburg paradox, and he concludes that the fair entry fee for playing N games of the St. Petersburg game is $\log{N}$ ...
1
vote
2answers
60 views
Why does the strong law of large numbers hold for the example of rolling a die?
I have read some good explanations on this site about the (weak and strong) laws of large numbers but I still have trouble applying the strong law of large numbers to the example of rolling a die.
I ...
1
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1answer
36 views
A result on power of random symmetric matrices and vector multiplication: u^T S^n v
I have a rough intuition about some results regarding random matrices theory, but I'm not sure if this is correct - if it exists already somewhere or if there is a way to prove it clearly.
Given:
$n,...
0
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1answer
22 views
Weakened presuppositions of the strong law of large numbers
Is there some version of the strong law of large numbers which only requires pairwise independent random variables, does not suppose identical distributions for them, but guarantees the convergence a. ...
1
vote
1answer
97 views
Using the law of large numbers to get a dominating sequence
Let $X_1, X_2, \dots $ be an iid sequence of random variables such that $EX_1 = \mu$, $E \vert X_1\vert^p < \infty$ for some $1<p<2$ and $E \log X_1 < 0$. The goal is to compute
\begin{...
0
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1answer
43 views
Prove that $\frac{-n}{\ln (\prod_{i=1}^n X_i)}$ converges to $\alpha$ almost surely
I'm solving the following exercise from previous final exam:
For $\alpha \in (0,1)$, we define the following density function $$f (x; \alpha) = \alpha x^{\alpha -1} \mathbf{1}_{[0,1]} (x)$$
We ...
0
votes
1answer
34 views
Law of Large Numbers (LLN) relation to Binomial Distribution
I am currently reading a book which explains the efficiency of Ensemble methods in statistical learning. In the following I modified the statements in that there is no need to understand anything ...
0
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1answer
58 views
In the Proof of Kolmogorov's Strong Law of Large Numbers
I understand everything in this proof concerning the strong law of large numbers, except for the line highlighted in red. I do not understand why $$\frac{X_1 + ...+X_n}{n}$$ is measurable with ...
0
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1answer
41 views
Show that a sequence satisfies the weak law of large numbers
Let $\{X_n\}_{n \geq 1}$ be a sequence of independent random variables such that $\mathbb P( X_n = \pm n^a)=1/2$.
Show that if $a<1/2$, then the sequence satisfies the weak law of large ...
2
votes
2answers
61 views
Is the following generalization of Strong Law of Large Numbers valid?
According to SLLN, if $X_1, X_2, \ldots$ is an infinite sequence of i.i.d. random variables with expected value $\mu$ and $S_n := \sum_{i=1}^n X_i/n$ then
$S_n \to \mu$ almost surely.
If the sequence ...
0
votes
2answers
54 views
Weak and Strong Law of Large Numbers
Let $(X_n)_n$ be a sequence of independent random variables such that:
$\mathbb P(X_n=1)=\mathbb P(X_n=-1)=\frac{1}{2}(1-2^{-n})$
$\mathbb P(X_n=2^{\frac{n}{2}})=\mathbb P(X_n=-2^{\frac{n}{2}})=2^{-...
2
votes
1answer
55 views
Prove $X_n$ fullfills Weak Law of Large Numbers
Use following facts:
$\forall_{\epsilon>0} \mathbb{P}(X \geq \epsilon) \leq \frac{\mathbb{E}X}{\epsilon}$ (Chebyshev's inequality)
When $\lim_n n \mathbb{P}(|X| > n) =0$ and $X_n^* = X \mathbb{...
2
votes
1answer
63 views
How to determine the sample size for a two sided $z$-test?
Let $X_{1}, \ldots, X_{n}$ be an iid sample from $N(\mu,\sigma^2)$
where $\sigma$ is known. We want to test a hypothesis
$$
H_{0}:\mu = \mu_{0}
\quad \mbox{versus} \quad
H_{1}: \mu \ne \mu_0
$$
Now, ...
0
votes
1answer
36 views
Adding an explicit upper bound in the Strong Law of Large Numbers
Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d variables with Bernoulli distribution $B(p)$. By the strong law of large numbers, we know that $\frac{X_1+\ldots+X_n}{n} \to p$ almost surely.
My ...
0
votes
1answer
40 views
Is Strong law of Large numbers equal to pointwise convergence?
1. function pointwise convergence
$\{f_n\}$ converges to $f$ pointwise on $E$ if the following equation holds,
$$f(x) = \lim_{n\to\infty}f_n(x) , $$
for every $x\in E$.
2. Strong Law of Large ...
-1
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1answer
49 views
Convergence of the expectation of a ratio of sample means
Suppose $\{(X_i, Y_i)\}_{i=1}^\infty$ is an iid sequence of random variables. Assume $EX_i^2$ and $EY_i^2$ are both finite and that $E(X_i) \neq 0$. Define
$$Z_n = \frac{n^{-1}\sum_{i=1}^n Y_i}{n^{-...
1
vote
2answers
30 views
Application WLLN in a convergence related problem
So, I have stumbled upon a question like this:
${X_n}_{\{n\geq1\}}$ be a sequence of iid random variables having common pdf $f_X(x)=xe^{-x}I_{x>0}$. Define $\overline{X_n}=\frac{1}{n}\sum_{i=1}^{...
1
vote
1answer
49 views
Law of large numbers for random variable
I am trying to understand the law of large numbers but maybe mixing some things up.
It says that for a sequence of random variables $X_1, X_2,...,X_n$ which are iid, then $n^{-1} \cdot \sum_1^\infty ...
1
vote
1answer
9 views
law of large numbers against several distributions
Suppose there are $K$ distributions $F_1,..,F_K$ and a random variable $x$ such that $E(x | F_i) = \mu_i$. Suppose also that the distributions appear in proportion $p_1 F_1 + ... + p_n F_K$, so that ...
3
votes
2answers
40 views
Law of large numbers on a random number of samples
Let $X_1,\dots,X_n$ be $n$ iid random variables, each with expectation $1$. From the law of large numbers one has
$$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i = 1 \text{ a.s.}$$
Now let $Y_n$ be a ...
2
votes
0answers
26 views
Feller's probability theory problem 5 from Ch. X - Conditions for LLN and CLT
I am Reading Feller's Introduction to Probability Theory and Its Applications and I found an exercise from Chapter 10 difficult to solve. In this exercise (which is the number 5 in the Intl. Edition) ...
0
votes
1answer
75 views
Prove that for any $\epsilon > 0$, $\mathbb{P}\left (\frac{S_n}{n} > p+\epsilon\right) \leq e^{-\frac{1}{4}n\epsilon^2}$.
$S_n = \sum\limits_{i=1}^{n} X_i$ where $X_i$'s are i.i.d RVs. $\mathbb{P}(X_i = 1) = p, \mathbb{P}(X_i = 0) = 1-p$. Prove that for any $\epsilon > 0$, $\mathbb{P} \left(\frac{S_n}{n} > p+\...
0
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0answers
22 views
Problem about central limit theorem related to finding probability of a R.V similar to binomial
In a city there is 1000 houses. We know that each house has 1/2 chance to not have any mouse. (So with 1/2 chance, it has 1 or more mice). A person gets 1$ for every ten mice he catches.
a) Find the ...
2
votes
0answers
50 views
Give an example of a sequence of zero-mean, finite variance random variables, with divergent sum of variance, and converging to $1$.
I am working on an exercise stated as follows:
Assuming $\sum_{k=1}^{\infty}p_{k}=\infty$, give an example of independent random variables $(X_{k})$ such that $Var(X_{k})\leq p_{k}<\infty$ and $\...
0
votes
1answer
26 views
An exercise where Kronecker's Lemma cannot be comfortably used.
I am working an exercise stated as follows:
Let $S_{n}=\sum_{k=1}^{n}Y_{k}$ for independent random variable $(Y_{k})$ such that $Var(Y_{k})<B<\infty$ and $\mathbb{E}Y_{k}=0$ for all $k$. Show ...
0
votes
0answers
23 views
Proof of Strong Law of Large numbers in Course of Prob Theory
I am trying to follow a proof of the SLLN in the textbook a course in probabilty theory.
Theorem: If the $X_j$ are uncorrelated and their second moments have a common bound then the SLLN follows:
...
3
votes
2answers
63 views
Law of large numbers and coin tosses
The law of large numbers is the following statement:
$\lim_{n \rightarrow\infty}(P(|S_n/n - \mu| < \epsilon))=1$, for all $\epsilon>0$
Where $S_n = X_1 + X_2 + ...+X_n$ iid random variables. ...
