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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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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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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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72 views

Is $\frac{1}{\frac{1}{n}\sum_{i=1}^{n}\frac{1}{X_{i}}}$ a consistent estimator for $\mu$

Let $X_i \sim N(\mu,\sigma^2)$, I want to find out if $$\frac{1}{\frac{1}{n}\sum_{i=1}^{n}\frac{1}{X_{i}}}$$ is a consistent estimator for $\mu$, or not. It's easy to show, using $LLN$, that $\...
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78 views

Can one construct a monotone law of large numbers?

let $X_1, X_2, \dots$ be a sequence of IID random variables defined on a probability space $( \Omega, F, P)$ with mean $E[X_1] = \mu $, define $$\bar{X}_n = \frac{1}{n}(X_1+ \dots+ X_n)$$ then $\bar{...
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493 views

Law of large numbers; Poisson distribution

Let $X_n$ be the numbers of job applications at a company in the year $1900+n,n\in\mathbb N$. Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent, identically distributed random variables with ...
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225 views

If a class of functions $\mathcal{F}$ is a Glivenko-Cantelli class then it is also a Donsker class?

Definitions: Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ with probability distribution $P$. All functions ...
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724 views

SLLN when the expectation in infinite

In a Post I found it says: Whenever ${\rm E}(X)$ exists (finite or infinite), the strong law of large numbers holds. That is, if $X_1,X_2,\ldots$ is a sequence of i.i.d. random variables with ...
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100 views

Law of Large Numbers contradicts Central Limit Theorem?

My text defines the weak law of large numbers: If $X_1,\ldots,X_n$ are IID, then $\overline{X} \overset{P}{\to} \mu$. And the CLT as: Let $X_1,\ldots,X_n$ be IID with mean $\mu$ and variance $\...
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A sequence for which the running average converges to something other than the expected value

$X_1,X_2,\ldots$ is a sequence of independent random variables and $$P\{X_n = n^2 - 1\} = 1 - P\{X_n = -1\} = n^{-2}$$ Clearly, $E[X_n] = 0$. However, $\frac{1}{n}S_n \to -1$ almost surely where $...
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How to prove/disprove this inequality?

Given a series of independent random variables $\{X_i\}$, such that: $P(X_i= i^{1/2})=\frac{1}{2i^{1/2}}$ , $P(X_i=-(i^{1/2}))=\frac{1}{2i^{1/2}}$, $P(X_i=0)=1-\frac{1}{i^{1/2}}$.i is a natural number....
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184 views

Probability ratio convergence in limit

The following is the question : Given random variables $X_1 \leq X_2 \leq \cdots$ such that $E[X_n] \sim A n^{\alpha}$, where $A,\alpha > 0$ ($\sim$ means that the ratio of the two quantities ...
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99 views

Rate of convergence for law of large numbers

You pass a street performer who offers you the following gambling deal: You have 1/3 chance of winning 3 USD and 2/3 chance of losing 2 USD. However you may only play one game. The street performer ...
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449 views

Law of large numbers and frequentist interpretation

The law of large numbers (WLLN) is a result that holds whatever be your interpretation of probability (i.e Bayesian, frequentist). You do not need to have an interpretation of probability to derive ...
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143 views

Random walk with centered increments

Let $X_1, X_2,\cdots$ be a sequence of independent, identically distributed random variables and $\displaystyle S_n=\sum_{i=1}^{n}X_i$. Then $EX_{1} <0$ if, and only if , $\displaystyle\lim_{n \...
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176 views

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have are given a distribution $p_{z}(k)$ over the whole $\mathbb{Z}^+$. We are interested in approximating $p_v(v)$ over ...
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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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Intuition behind almost sure limit of $\frac{|S_n|}{n^{1/p}}$

Suppose $X_i$'s are non-degenerate i.i.d. Then (1) If $E|X_1|^p=\infty$ we've $\limsup_{n\rightarrow \infty} \frac{|S_n|}{n^{1/p}}=\infty$. And this is true $\forall p>0$ (2) However for $p=2$ ...
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The law of large numbers - limits of $\max$ vs $\max$ of a limit.

Assume that $X_{1,1}, \dots , X_{1,n}, X_{2,1},\dots, X_{2,n}, \dots ,X_{n,1}, \dots , X_{n,n}$ are i.i.d. random variables, and that $\mathbb EX_{i,j}$ exists and is finite. From the strong law of ...
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98 views

empirical estimation of Bernoulli distribution (lower bound)

Let $X_i$ be an i.i.d. Bernoulli distributed sequence, with probability $p$ being 1. Now consider an empirical estimation of $p$ with $l$ samples and I am looking for a lower bound for following ...
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145 views

Almost sure limit of $\log(X_1 + X_2 + … + X_n) - \log(n)$

Let $X_n$ be an i.i.d. sequence of positive random variables with expectation 2 and variance 1. What is the almost sure limit of $$\log(X_1 + X_2 + ... + X_n) - \log(n)$$ as $n \to \infty$ Would it ...
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157 views

Sum of sequence of random variables infinitely often positive

Let $X_1,X_2,\ldots$ be an infinite sequence of independent (but not necessarily identically distributed) random variables with $E(X_i)=0$ for all $i$. Set $S_n=\sum_{i=1}^n X_i$. I want to show that ...
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105 views

simple proof of the $L^2$ weak law for discrete-time ergodic Markov processes

Let $\{X_t\}_{t\in\mathbb{Z}}$ be a stationary and ergodic stochastic process with finite second moment. Von Neuman's ergodic theorem implies that the time average $(1/N)\sum_{j=0}^{N-1} X_j$ ...
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240 views

Relation between Strong and Weak Law of Large Numbers

I am trying to prove the following theorem somewhat indicating the relationship between Strong and Weak LLNs: Let $\{S_n\}$ be the partial sums of a series of independent r.v.'s $\{X_n\}$. Then $...
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158 views

SLLN of Markov chains .

Let $X_1$, $X_2$,... be a finite state, irreducible and aperiodic Markov chain with initial state $X_0=i$. It is known that \begin{equation} \mathrm{P}\Big\{\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{...
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179 views

random walk with possibility to freeze

Consider a Random Walk on a one-dimensional lattice. The walker starts moving at time $0$ from $x=0$. At every step, the walker moves to the right with probability $p$, to the left with probability $q$...
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What is the difference between ergodicity and the law of large numbers?

I want to begin by saying that I know absolutely no measure theory. To my knowledge, roughly speaking a stochastic process is ergodic if its time average converges to the expectation (space average) ...
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A family of continuous distribution functions with a singular law to the Lebesgue measure

Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $\{X_n: n \ge 1\}$ be a sequence of i.i.d random variables with the common distribution $$\mathbb{P}(X_1 = 1) = p \text{ and } \...
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Variants of weak and strong LLN

Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $\{X_n: n\ge 1\}$ be sequence of square integrable random variables, i.e., $X_n \in L^2(\Omega, \mathcal{F}, \mathbb{P})$ for each $...
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47 views

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 ...
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Convergence in prob. and a.s.: Commonly-used strategies

Good afternoon, I'm typing in $\LaTeX$ a formularium and a compendium of useful probability theorems, corolaries and commonly-used approaches to many topics in preparation to an important exam at my ...
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47 views

When does SLLN hold for specified characteristic function?

Let $X_i$ be sequence of i.i.d. random variables. Suppose $0 < \alpha \leq 2$ and characteristic function is given by \begin{equation} \varphi(t) = e^{-|t|^\alpha}\quad \left[{}=\exp(-|t|^\alpha)\...
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111 views

Sum of Independent Poisson Distribution

Let $X_k$ be sequence of independent random variables and suppose $X_k$ has Poisson distribution with mean $k^r$. For which values of $r$, \begin{equation} \lim_{n\to\infty}\frac{\sum_{k=1}^n X_k}{\...
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46 views

Is this rule a consistent estimator?

Define the sample mean $S_n=(X_1+...+X_n)/n$ of a sequence $\{X_1,...,X_n,...\}$ of i.i.d. random variables with non-negative discrete support, known mean $0<\mu<1$, finite standard deviation $\...
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41 views

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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74 views

Using the WLLN to find the limiting distribution of a random variable

I have a random variable $X_n$ that arises from the following random walk on $\mathbb{Z}^n$: let $X_0 = 0$ and $\mathbf{P}(X_{n+1} = X_n + 1 \mid X_n) = p$ and $\mathbf{P}(X_{n+1} = X_n - 1 \mid X_n) =...
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50 views

Limit of AM/GM ratio for large collections of numbers

I have encountered an interesting statement related to the law of large numbers. Namely if any function of $n$ variables has finite gradient for $n \to \infty$ then it is asymptotically constant. (I ...
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243 views

i.i.d. sequences satisfies the weak law but not strong law

Is there an example of an i.i.d. sequence of random variables $X_n$ so that $S_n/n\to 0$ in probability but $S_n/n$ does not converge a.s.? This post has an answer in which the sequences are ...
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43 views

Law of large numbers and cardinality of intersection

Let sets $A,B$ s.t. $A\subseteq B$ with cardinality $|A|=\frac{q}{n}$ for integers $q,n$ and $q$ a multiple of $n$. Assume that the sample space is finite and let's consider an element $x$. In the ...
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A question on WLLN: Does such $b_n$ always exists?

Question: Let, $b_n=\inf\{ y\geq 1: n\mu(y)/y\leq 1\}$ $$\int_{0}^{b_n}\mu^2(y)dy\leq \int_{0}^{b_n}\frac{y^2}{n^2}dy=\frac{b^3_n}{3n^2}$$ Now, $nP(|X_1|>b_n)=nP(X_n>b_n)=n[1-F(b_n)]=\dfrac{n}{...
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41 views

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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94 views

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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224 views

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 $\mathbf{p}=\{p_1,p_2,\dots,p_m\}$...
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454 views

Sequence of random variables, mean zero, convergence to -infinity

What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ?$$...
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170 views

Central limit theorem for uncorrelated identically distributed random variable

I have a sum of random variables as bellow $$Y=\sum_n A_n=\sum_n B_n\times C_n$$ where $B_n$s are correlated Gaussian random variables with zero mean, variance $1$ and correlation $E\{B_nB^*_r\}=\frac{...
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77 views

Bounds on probability of sample mean in a small neighborhood rather than the tail

Say we have i.i.d random variables $x_i$ whose mean and variance are $1$. Then the sample $s_n=\frac{1}{n}\sum_{i=1}^n x_i$ has mean $1$ and variance $\frac{1}{n}$. If we are given a small enough ...
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46 views

Interpretation of different convergence results for random series

Let $X_k, k\geq 1$ be a sequence of random variables and let $S_n:=\sum_{k=1}^n X_k, n \geq 1$ be the sequence of partial sums. When the $X_k$ are Independent, Kolmogorov's 3-series Theorem gives ...
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203 views

Limit of cdf of binomial distribution

I would like to compute the limit of CDF for a Binomial distribution as $n \rightarrow \infty$, \begin{equation*} \lim_{n \rightarrow \infty} F( \theta;n,q) = \lim_{n \rightarrow \infty }\sum_{k=0}^{...
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120 views

Convergence of Sample Mean Via WLLN

I am trying to show that the sample variance converges to the population variance in using the Weak Law of Large Numbers $$\begin{align} \\ \Rightarrow S_n= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2 &...
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100 views

Markov chain and laws of large number

Let $X_n$ be a (could assume this is homogenous) markov chain on a countable state space $S$, and write $N_n(x)=\sum_{k=1}^n 1_{\{X_k=x\}}$. Let $z\in S$ be a recurrent state, denote by $R_z$ its ...
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0answers
66 views

Conditions for convergence to Gaussian distribution

Let $$n_i(t)= H(u_i(t))$$ where $N\geq i\geq 1$, $H(.)$ is the Heaviside function and $$ u_i(t) = \sum_{j=1}^N J_{ij} n_j(t) $$ We start with a random $\vec{n}(0)$ and each step of time $t'<t$...