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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11 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 ...
2
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1answer
14 views

Maximizing the probability of a poll prediction

Using the central limit theorem, I was able to find out the first part of this question. However, part b is eluding me. How do I, in general, find a value for $n$ such that we can ensure the ...
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0answers
35 views

$ \lim\limits_{n\rightarrow \infty} {\frac{B_{t}}{t}}$ Brownian Motion

I want to prove in two ways that $ \lim\limits_{n\rightarrow \infty} {\frac{B_{t}}{t}}\rightarrow 0$ almost surely, where $B_{t}$ is a standard Brownian Motion. 1) in $L^2$ Can we say ...
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1answer
14 views

Reformulation of SLLN with continuous nondecreasing process as time

Given a filtered probability space $(\Omega,\mathcal{F}_{t},P)$ and a continuous nondecreasing process $U_{t}$ with $U_{0}=0$ and $U_{t}\rightarrow \infty$ $P-a.s.$ as $t$ goes to $\infty$. Given a ...
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1answer
23 views

Convergence of product of random variables with distribution $U(0, e)$

Let $X_1, X_2, \ldots$ be a sequence of independent random variables with uniform distribution on $[0, e]$. Let $R_n:=\prod_{k=1}^n X_k$. By Kolmogorov's zero–one law $(R_n)$ converges with ...
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0answers
23 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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1answer
32 views

How do I transform this Probability in weak law of large number form?

Let $X_{1},\dots$ be a sequence of independent random variables. Suppose, for $k=1,2,\dots$ $$P\left(X_{2k-1}=1\right)=P\left(X_{2k-1}=-1\right)=\frac{1}{2}$$ and the probability density function of ...
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0answers
24 views

Test a law-of-iterated-logarithm-like result, with numerical simulation

I have a non-standard random walk $S_n$ for which the increments are not exactly independent (I could describe it, but it would be a totally different long and complex topic). I expect it to have ...
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0answers
24 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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3answers
38 views

Convergence of independent $\mathcal U {(n,n^2)}$ random variables?

What does this sequence of random variable's distribution converge to? The random variables are given as follows $$Y_n=\frac{X_n-n}{n^2}, \quad n=1,2,3, \dots$$ and $X_n\sim\mathcal{U(n,n^2)}$- a ...
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0answers
53 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 ...
2
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1answer
51 views

Is there a law of large number for weighted sum of independent random variables

I have the following sequence $\{Y_n\}$ of random variables defined as $$Y_n=\frac{\sum_{k=1}^{n}a^{k-1}X_k}{\sum_{k=1}^{n}a^{k-1}}$$ where $\{X_n\}$ is a sequence of i.i.d. random variables. My ...
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1answer
32 views

Proving Borel Strong Law of Large Numbers using Bienaymé-Tchebichev inequality.

While reading Loeve's book on Probability (page 246, 5th edition) I found the following statement: It is of some interest to observe that Borel's law of large numbers can also be obtained by ...
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1answer
41 views

Weak Law of large numbers involving a sequence and random variable

During one of our Information theory classes, the professor constructed the following set: $$T_\delta = \left\{\mathbf{y} \in \mathbb{R}^n: \frac{\sum_{i=1}^ny_i^2}{n} \leq P + ...
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0answers
25 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 ...
2
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0answers
50 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 ...
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1answer
35 views

If $X_1,\ldots X_n$ are iid random variables with $E(X_{i}) = 0$ and $E(X_{i}^2) = 1$, then $\frac{\sum_{i=1}^{n}X_i^2}{n} \to E(X_i^2)$?

I am trying to see why it is that if $X_1,\ldots X_n$ are i.i.d. random variables with $E(X_{i}) = 0$ and $E(X_{i}^2) = 1$, then by the Law of Large Numbers we have that $$ ...
2
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1answer
32 views

Show that law of large numbers hold

Given,Xi's are iid random variables and $$f(x)= \frac{1+δ}{x^{2+δ}}$$ $δ>0$ and $X>1$ To show that law of larger numbers hold, I used khinchin's theorem which states that if Xi's are iid then ...
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1answer
38 views

Strong Law of Large Numbers converge a.s.

The Question is: Let ($X_{n})_{n≥1}$ be a sequence of i.i.d. Bernoulli random variables, on the same probability space, with parameter $\frac{1}2$ (P($X_{n}$ = 0) = P($X_{n}$ = 1) = ...
2
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0answers
26 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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36 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 ...
2
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1answer
41 views

Probability exercise about SLLN

The question: Let {$X_n$} be i.i.d random variables. $EX_1=0$. Then $\sum_{i=1}^{n}X_i\over n$ converges almost surely to zero. I know that when the sequence $\{X_n\}$ satisfies ...
2
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1answer
86 views

How to apply law of large numbers for this problem

Let $X_1,...,X_n,...$ be independent variable satisfying $P(X_i=0)=P(X_i=1)=\frac{1}{2}$ for all i then denote $Z_i=X_iX_{i+1}$ for all i .I want to show that $\lim_{n\to ...
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1answer
47 views

Comparing Monte Carlo estmated PI and the real value PI

A famous example of Monte Carlo integration is the Monte Carlo estimate of PI. The unit disk { (x, y) : x2 +y2 <= 1 } is inscribed in the square [ 1, 1] x [ 1, 1], which has area 4. If we ...
2
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0answers
53 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 ...
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0answers
18 views

convergence of ratio of random variables

$\overline{X}\overset{p}{\to} 0$, $\overline{Y}\overset{p}{\to} 1$ where $\overline{X}=\frac{1}{N}\sum X_i$ and $\overline{Y}=\frac{1}{N}\sum Y_i$. Also $X_i\sim iid$ , $Y_i \sim iid$ but $X_i ...
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1answer
70 views

Using Borel-Cantelli Lemma to show the almost sure divergence of $S_n/n$

We have independent random variables such that $$\mathbb{P}(X_n=n)=\mathbb{P}(X_n=-n)=\frac{1}{2(n+1)\ln(n+1)}$$ and $$\mathbb{P}(X_n=0)=1-\frac{1}{(n+1)\ln(n+1)}$$ I am trying to show that ...
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0answers
36 views

Law of large number for a product of uniform iid random variables (stick breaking)

Let $(X_n)_n$, $n = 1, 2, \dots$, be an iid sequence of random variables uniformly distributed on $(0, 1]$. Set $S_0 = 1$ a.s. and, for $n = 1, 2, \dots$, set $S_n = \prod_{k=1}^n X_k$. Compare $S_n$ ...
2
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1answer
41 views

How can one find $\mu$?

Let $X_n$ be iid random variables with $P(X_n=(-1)^kk)=\dfrac{C}{k^2\log k}$ for all $k\geq 2$. Here $C$ is just a constant so that the sum of probabilities is $1$. Find constant $\mu$ such that ...
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1answer
15 views

limit of average of independent, but not identically distributed r.v.

Let $\{X_i\}$ be a collection of independent r.v., but with distribution dependent on index $i$, such that $P(X_i=2^i)=2^{-i}$ and $P(X_i=0)=1-2^{-i}$ for $i \in \mathbb{N}$. What can I say about ...
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1answer
41 views

Proving almost sure convergence

Assume the sequence of random variables $X_1, X_2, \cdots$ are IID with finite mean and finite variance. Define a random variable: \begin{align} Y_n = \frac{X_n}{n} \end{align} Show that $Y_n \to 0$ ...
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29 views

Showing a sequence converges in probability

I'm studying for a test on Monday and am going through some supplementary problems. These problems do not come with solutions provided but I still think they are very good practice. Suppose sequence ...
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0answers
27 views

If a sequence of Poisson RV's converges to $0$ show their parameters converge to $0$

Let $X_n$ be a sequence of random variables such that $X_i \sim Poi(\lambda_i)$. If $X_n$ converges to $0$ almost surely show that $\lim_n \lambda_n = 0$. If $X_n$ converges to $0$ almost surely, ...
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1answer
32 views

Does $Z_n=\sum_{k=1}^{n}\sqrt{k}X_k$ satisfy the strong law of large numbers if $ X_n…$

Does $Z_n=\sum_{k=1}^{n}\sqrt{k}X_k$ satisfy the strong law of large numbers if $ X_n: \begin{matrix}-\frac{1}{n} & \frac{1}{n} \\ \frac{1}{2} & \frac{1}{2} \end{matrix}, n=1,2,...$ are ...
2
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1answer
49 views

Convergence in probability using characteristic functions

I need to solve the following problem. Let $X_1,X_2,\dots$ be independent random variables all with expectation $0$ and variance bounded by $M$. Prove that $\frac{1}{n}\cdot \sum\limits_{k=1}^{n} ...
2
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1answer
26 views

Proving convergence with probability 1

We have a sequence $X_1, X_2, \cdots$ of IID R.V's. We are also given that $E[\log(X_i)]$ exists. We define a new sequence of random variables in terms of the $X_i$'s: $Y_n = (X_1X_2\cdots ...
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25 views

Strong law of large numbers and the probability that something exists.

The strong law of large numbers states that if $X$ is a RV and $X_1, X_2 \ldots$ are independent and identically distributed copies of $X$, and $\overline{X}_n= \frac{1}{n}(X_1 + \ldots + X_n)$, and ...
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1answer
42 views

What does it mean to “Converge in Law”?

If $X_1, X_2, X_3, \cdots$ is a sequence of independent identically distributed random variables with $E[X_i] < \infty$ and $Var(X_i)< \infty$ such that the sequence $Y_n = 3 \frac{X_1 + X_2 + ...
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2answers
160 views

Measuring $\pi$ by throwing darts

I want to give an approximation of $\pi$ in this way: I inscribe a circle in a square then I throw darts at random on the square from far away. If the darts falling on the square are $n$ and the ...
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2answers
91 views

Questions on Inverting Laplace transforms and Probability

From Williams' Probability w/ Martingales: Are we allowed to switch derivative and integral as follows: $$\frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x} f(x) = ...
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0answers
54 views

Limiting sequence of exponential random variables

Let $\eta_k$ be i.i.d. random variables having an exponential distribution, $$F_\lambda(x) = P(\eta_k \leq x) = 1-e^{-\lambda x}$$ for $x \geq 0$. Consider a sequence $\xi_k = ...
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1answer
60 views

First version of strong law / $\lim a_n^4 = 0 \to \lim a_n = 0$?

From Williams' Probability w/ Martingales: How does the conclusion follow? Guess 1: $E[\sum (\frac{S_n}{n})^4] < \infty$ $\to \sum (\frac{S_n}{n})^4 < \infty$ a.s. $\to \lim_{n \to ...
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1answer
27 views

Prove inequality in first version of strong law

From Williams' Probability w/ Martingales: How exactly does that inequality hold true? I get that $E[X_i^2] \le E[X_i^2]^2 \le K$ and $E[X_j^2] \le E[X_j^2]^2 \le K$, but how does that mean ...
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1answer
70 views

monte carlo simulation - confidence intervals construction

I am starting with Monte Carlo Simulation. I have run simulation to estimate the mean and the variance of the exponential distribution. Simulation: I have generated random sample from uniform ...
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0answers
18 views

Step in proof that if $\Bbb{E}(X_n)\to\mu$ and Var$(X_n)\to0$ then $X_n \overset{P}{\to} \mu$

Let $(X_n)_{n\geq 1}$ be a sequence of random variables such that $\lim_{n \to \infty}Var(X_n)=0$. Show that if $\lim_{n \to \infty}\Bbb{E}(X_n)=\mu \in \Bbb{R}$ then $X_n \overset{P}{\to} ...
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1answer
40 views

Probability problem - Converse to SLLN

I'm having trouble with an exercise (E4.6 Converse to SLLN) in "Probability with Martingales" by David Williams. The problem is as follows: Let $Z$ be a non-negative RV. Let $Y$ be the integer part ...
2
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2answers
42 views

Limits- legal and illegal algebraic manipulation.

I am doing probability and am using the strong law of large numbers to get after a bit of irrelevant extra algebra which I wont mention $\frac{\log(C_n)}{n} \rightarrow K$ where $C_i$ are a sequence ...
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1answer
26 views

Convergence of random variables - LLN (proof)

I need help. Prove: For arbitrary $\left\{ X_{n}\right\} $, if $\sum_{n}\mathbb{E}\left[X_{n}\right]<\infty$ then $\sum_{n}X_{n}<\infty$ converge absolutely almost everywhere.
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0answers
24 views

weak law of large numbers, exponential(3) distribution

I'm going through an example in my textbook but can't seem to follow every step. Let $ W_1, W_2,... $ be i.i.d with distribution $ Exponential(3). $ Prove that for some $n$, we have $ ...
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0answers
48 views

Law of large numbers - different definition of expected value

Looking at the proof of the Law of large numbers, you can tell it doesn't refer directly to the definition of expected value. I know it's wrong to assume the LLN would hold with a different ...