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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).

The law of large number (lln) describes what happens if we perform an experiment a large number of times. It states that the average of the results obtained from a lot of trials will be close to the expected value. It also states that it will get closer to the expected value as more trials take place. The law guarantees a stable long-term results for the averages of events.

The strong law of large numbers states that the averages converge a.s., to the expected value \begin{equation*} \overline{X}_n\to \mu,~ n\to\infty . \end{equation*}

The weak law of large numbers states that the average converges in probability towards the expected value: \begin{equation*} \lim_{n\to\infty}Pr(|\overline{X}_n-\mu|>\epsilon)=0. \end{equation*}

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Difference in conditions of Weak and Strong Law of Large Numbers

I understood the difference between convergence in probability and almost surely. And after searching online, I found multiple counterexamples to show that the WLLN does not necessarily imply the SLLN. However I am still failing to find the…
Dylan Zammit
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Limit of $\frac 1 {\sqrt n} \int_0^{1} \int_0^{1} ...\int_0^{1} \sqrt {\sum_{i=1}^{n} x_i^{2}} dx_1...dx_n$

What is limit of $\frac 1 {\sqrt n} \int_0^{1} \int_0^{1} ...\int_0^{1} \sqrt {\sum_{i=1}^{n} x_i^{2}} dx_1...dx_n$ as $n \to \infty$? This question was just deleted probably because there were errors in the statement. So I am posting it and…
Kavi Rama Murthy
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How can the Law of Large Numbers give approximations of the expected value?

Consider $10000 \ N(0, 1)$-distributed r.v.’s and let $Z_n = \frac{1}{n} \sum_1^n X_i^2$ Calculate $E(X^2_1 )$. How does this relate to the statement of the Law of Large Numbers? Explain how this can be used to give approximations for $E(g(X))$…
user9750060
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What is the difference between weak and strong laws of large number?

I've read the few posts on SE about weak vs strong law of large numbers, but I still can't quite differentiate the 2. Mathematically, it looks like the limit is applied to the probability, whereas in the weak law the limit is applied to the…
user5965026
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Law of large numbers with mean $\mu$ depending on $n$

Let $X = (X_i: i=1,\ldots,n)$ where $X_i \sim f$ i.i.d. such that $f$ is supported on $[0,\infty)$, $E(|X_i|)<\infty$ and $E(X_i)=\mu$. Consider $\tilde{X} = (X_i: X_i\leq a_n)$ for some deterministic sequence $a_n \to \infty$ and denote $m$ the…
Gohan
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Limitation of law of large numbers

The book "Statistical Learning Theory" by Vladimir Vapnik has a part which I cannot understand properly. "According to the classical law of large numbers, the frequency of any event converges to the probability of this event with an increasing…
nevernight
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Is there an "inverse law of large numbers"?

The different laws of large numbers that I've seen state that If conditions $K$ holds Then the sample average of a process X_n converges in probability/almost-surely, to $\mu$. Is there an inverse of this? If conditions $L$ do not hold Then…
user56834
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Rate of convergence for 'Law of large numbers'

Consider the following question: A coin has the probability of landing of head equal to 1/4 and is flipped 2000 times. Use the law of large numbers, find a lower bound to the probability that the total number of heads lies between 480 and…
Steel
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Upper bound on the weak law of large numbers in the opposite direction

We known that for i.i.d. RVs $X_i$, where $i=1,2,...,n$, the following holds $\Pr\Big(\big|\frac{1}{n}\sum_{i=1}^n X_i -E[X]\big|<\epsilon\Big)\geq 1 - \frac{\sigma_X^2}{n \epsilon^2}$. However, is it possible to obtain an upper bound on the above…
Nik
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SLLN for iid $X_n$

For i.i.d $X_n$'s with $E[(X_1)_-]<+\infty$ and $E[(X_1)_+]=+\infty$, I want to prove that $$\frac{1}{n}\Sigma_{i=1}^nX_i\xrightarrow{\text{a.e}} +\infty$$ as $n\xrightarrow{}+\infty$. I know that I have to use the Strong Law of Large numbers but I…
statwoman
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Law of large numbers about non identical random sequence.

I want to show the following statement. Statement : $Z \to 0 $, when $N \to \infty $. where $Z = \frac{{\left| {{e^{j{\theta _1}}}\left| {{x_1}} \right| + {e^{j{\theta _2}}}\left| {{x_2}} \right| + ,..., + {e^{j{\theta _N}}}\left| {{x_N}} \right|}…
rasidz
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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 with probability $p_i$, the distribution faced is…
user341502
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Law of large numbers for a sequence of independent random variables with non-converging parameter

There are a number of questions here about the applicability of the LLN to a sequence of independent Bernoulli random variables $X_n \sim B(p_n)$ when $p_n \to p$. What happens if the sequence $p_n$ does not converge to any $p$? It seems to me that…
Confounded
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Strong law of large numbers for Bessel process

I'm trying to prove that $$\lim_{t\to\infty}\frac{R_t^{(\nu)}}{t}=0 \quad \text{ a.s}$$ where $R_t^{(\nu)}$ is a Bessel process of order $\nu$. According to my source we obtain this directly from knowing that $$R^{(\nu)}\sim\{Z_t:t\geq0\}$$ where…
a.s
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Explain the following behavior

I generated a list of 8000000 random numbers in python between 1 and 100000000000000. import random random_numbers = random.sample(range(1, 100000000000000), 8000000) #Initialize empty list of random zeron elements counters = [0] * 10 for item in…
J.Doe
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