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Questions tagged [probability-limit-theorems]

For question about limit theorems of probability theory, like the law of large numbers, central limit theorem or the law of iterated logarithm.

This tag should not be used for questions about deterministic limits.

For question about limit theorems of probability theory, like the law of large numbers, central limit theorem or the law of iterated logarithm.

1553 questions
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Coupon collector problem convergence

For a positive integer $n$, there are $n$ different coupons, and you are trying to collect them all. Each time you purchase an item, you receive one of the $n$ coupons uniformly at random. Let $T_n$ denote the amount of time it takes to…
Anonymous
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Simple counterexample for law of large numbers and central limit theorem

I'm looking for a sequence of identically distributed random variables $(X_n)_n$ with $\text{var} (X_i) > 0$ so that the law of large numbers (either weak or strong) and the central limit theorem do not apply. I propose: $\mathbb{P} [X_1 = 0] =…
Andy
  • 261
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Convergence in Distribution implies convergences almost surely

Suppose we have a sequence of non decreasing random variables that converges to 0 in distribution. How can we prove that this sequence converges to 0 almost surely?
user321821
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Prove the modification of the Central Limit Theorem

I know we need to use some modification of the Central Limit Theorem here. But we can't use it directly because $(x_i)$ are not independent. I tried to substitute the representation of $x_i$ but I don't know what to do next. Please help me with…
Jennifer
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Is the derivative of the expected value equal to the expected value of the derivative

Is the derivative of the expected value equal to the expected value of the derivative of a random function? That is, is $\partial E[f(.)]/\partial x = E[\partial f(.)/\partial x]$?
user187975
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Clarifying a Step in Central Limit Theorem Derivation

So I'm trying to understand how to derive the central limit theorem but I'm confused about step 3 in Peter Young's Derivation In step 3, rightmost part of the equation, why are we taking the sum of all $x_i$, then subtracting them by $X$, which is…
Iancovici
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$(1-F(y))y \rightarrow 1$ as $y \rightarrow \infty$

I am trying to understand the proof given in Appendix A of https://arxiv.org/pdf/2103.00083.pdf. At the final part, the authors are proving that $(1-F(y))y \rightarrow 1$ as $y \rightarrow \infty$, where $F(y)$ is a cumulative probability…
user1125373
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Proving an Absolute Value Inequality

I've just started reading Spivak's Calculus text (4th ed.) and am having some trouble on one of the exercises. The problem asks me to prove that if $|x-x_0|<\frac{\epsilon}{2}$ and $|y-y_0|<\frac{\epsilon}{2}$, then $|x-y-(x_0-y_0)|< \epsilon$. I've…
James Pirlman
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Estimate of binomial distribution with its expectation

This exercise is a direct proof of the Law of Large for the Bernoulli case. Let $\xi_1,\cdots,\xi_n$ be i.i.d. r.v. with $P(\xi_k=1)=p=1-q=1-P(\xi_k=0)$. Prove that…
Xiong Jiangnan
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Almost certain non-existence of limit

A conjecture in need of a proof Let $P$ be a probability distribution (satisfying countable additivity) with outcome space $\mathbb{N}=\{1,2,3,\ldots\}$. Let $N$ be a stochastic variable $\mathbb{N}\to\mathbb{N}$. For each infinite sequence of…
Casper
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Convergence in total variation implies convergence of entropy?

Say I have a sequence of probability measures $\mu_n$ with Lebesgue densities $f_n$ that converges in total variation norm to a probability measure $\mu$ with Lebesgue density $f$. Further assume that $f$ and $f_n$ are continuous and bounded, and…
J. Curwen
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If $X_n=O_P(1)$ and $Y_n=o_P(1)$, prove $X_nY_n=o_P(1)$

We say $X_n=O_P(1)$ if $X_n$ is bounded in probability. We say $Y_n=o_P(1)$ if $Y_n$ converges in probability to $0$. My attempt: Since $X_n=O_P(1)$ and $Y_n=o_P(1)$, we have $$ \forall\ \epsilon>0,\ \exists\ M \text{ and }\ n_0 \text{ such that }…
Xiangdong Meng
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convergence of average of random variables

Say $X_i$ are iid random variables sampled from a uniform distribution on the real line $\mathbb{R}$. (1/n)$\Sigma_{i=1}^n$ $X_i$ "converges" to $\int_{\mathbb{R}}$ xp(x)*dx where p(x) is the probability density function for the normal distribution.…
user352102
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Multivariate Central Limit Theorem:

I am reading a book which talk about MCLT, in which they give one result that I don't understand. Let A=$\sum _{s=1}^{n}x_sx'_s$, where $x'$ denote the transpose of $x$, $x_s=(x_{s(1)},...,x_{s(p)})$ is a p-variate random vector so that $x_s\sim…
user229013
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Basic: Sequence of random variables and CLT

I've started reading about Limit Theorems in Probabilities.The chapter starts by saying, let's think of a sequence $S_n = X_1+X_2+X_3$ iid random variables. What is each random variable in the sequence. Is it a value from each r.v. or all the…
user1011001229
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