# 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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### Tail Probabilities of a martingale difference sequence

I'm currently facing the problem, that I can neither prove nor find a counterexample for the following statement. Let $q \in (1,2)$ and let $(D_n)_{n \in \mathbb{N}}$ be a martingale difference ...
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### Gap between two consecutive order statistics under arbitrary distribution.

Consider an arbitrary distribution $\mathcal{D}$ supported on $[a,b]$ with density function $\phi(x)\in[\gamma, \Gamma]$ where $\Gamma\geq \gamma>0$. M i.i.d samples $\{d_i\}_{i=1}^M$ are drawn ...
3 votes
1 answer
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### Law of large numbers with incomplete observation

I am currently reading the book Introduction to Reinforcement Learning by R. S. Sutton and A. G. Barto. The authors often reason with the LLN. In particular, at one point there is an expression like ...
2 votes
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1 vote
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### The Equivalent Condition of the Weak Law of Large Numbers When Random Variables Are Uniformly Bounded

When the random variable $\{X_n,n\ge1\}$ satisfies the uniformly bounded condition, why does $$\frac{1}{n^2}\operatorname{Var}\left(\sum_{k=1}^{n}X_k\right)\rightarrow0$$ become a necessary and ...
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1 answer
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### Proving $\frac{1}{n}(\sum_{k=1}^{n}a_k\xi_{k}+\sum_{k=1}^{n}\eta_{k})\stackrel{P}\to 0 \iff \frac{1}{n^2}\sum_{k=1}^{n}a_k^2\to0$

A problem related to the weak law of large numbers，$\{\xi_k\}$ and $\{\eta_k\}$ are independent of each other and $\{\xi_k\}$ and $\{\eta_k\}$ are all independent sequence. They all obey the standard ...
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1 vote
1 answer
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1 vote
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### Bounded first moment vs. bounded absolute moment?

I notice a couple different versions of the law of large numbers, one that assumes finite mean, another that assumes finite absolute first moment. Are these equivalent? I think we can argue one ...
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### What happens when you add n Cauchy random variables that have a positive correlation?

The reason the law of large numbers fails to apply for the Cauchy distribution is that the distribution of $X_1+X_2+X_3+\dots X_n$ is the same as $X_1+X_1+\dots X_1 = n X_1$. This is billed a curious ...
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1 vote
1 answer
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### Law of Large Numbers and Convergence in Probability of Quadratic Form

Let $\{\mu_N\}_N$ be a sequence of $N$ dimensional vectors, and $\{\Lambda_N \}$ a sequence of $N \times N$ symmetric positive definite precision matrices. I drop the $N$ subscripts for simplicity of ...
-1 votes
1 answer
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### Brownian motion running maximum [closed]

Could someone please explain how to find the limit as $t \rightarrow \infty$ of the running maximum of a brownian motion $B_t =\max W_t-ut$? Is there a way to calculate the limit itself and not just ...