# 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).

186 questions
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### Suppose $E[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s.

Let $X_1,X_2,X_3,...$ be i.i.d. with $P(X_1 >0)=1$. Define $S_n =\Sigma_{i=1}^{n} X_i$. (a) Suppose $\mathbb{E}[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s. I ...
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### Finding the expected value and variance for $Y_n^{(c)}$

I am to find all $\beta > 0$ such that the following series converges: $$\sum \limits_{n = 1}^{\infty} n^{- \beta} \big(X_n - E(X_n) \big). \tag{1}$$ $X_n$ is a random variable with exponential ...
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### Growth of brownian paths (Loose bound)

I'm trying to figure out a statement from page 166 of the book "Brownian Motion: An introduction to stochastic processes" by Schilling, Partzsch (2012). The chapters illustrates how fast the brownian ...
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### Convergence of empirical distribution function of a sequence from dependent identically distributed rvs

We know by the Law of Large Numbers that the empirical distribution function converges to the common distribution of an iid sequence. What happens if we drop the independence requirement. Will the ...
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### Sum of correlated random variables and the Law of Large Numbers?

Suppose I have a random variable $X$ which can take values on the set $\mathcal{X}=\{1,2,\dots,m\}$ and $X$ is drawn according to the given probability mass function $\mathbf{p}=\{p_1,p_2,\dots,p_m\}$...
What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ?$$...