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.

1304 questions
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Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: Let $(X_n)$ be a sequence of dependent…
27
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3 answers

Probability and measure theory

I'd like to have a correct general understanding of the importance of measure theory in probability theory. For now, it seems like mathematicians work with the notion of probability measure and prove theorems, because it automacially makes the…
23
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2 answers

Convergence of series $\sum\limits_{k=1}^\infty\frac{1}{X_1+\dots+X_k}$ with $(X_k)$ i.i.d. non integrable

Pick a sequence $X_1$, $X_2$, $\dots$, of i.i.d. random variables taking values in positive integers with $\mathbb{P}(X_i=n)=\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}$ for every positive integer $n$. Q: Does the sum…
22
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How often was the most frequent coupon chosen?

In the coupon collector's problem, let $T_n$ denote the time of completion for a collection of $n$ coupons. At time $T_n$, each coupon $k$ has been collected $C_k^{n}\geqslant 1$ times. Consider how often the most frequently chosen coupon, was…
Did
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Rate of convergence in the central limit theorem (Lindeberg–Lévy)

There are similar posts to this one on stackexchange but none of those seem to actually answer my questions. So consider the CLT in the most common form. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables with $X_1 \in L^2(P)$…
16
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2 answers

Prove that the maximum of $n$ independent standard normal random variables, is asymptotically equivalent to $\sqrt{2\log n}$ almost surely.

Lets $(X_n)_{n\in\mathbb{N}}$ be an iid sequence of standard normal random variables. Define $$M_n=\max_{1\leq i\leq n} X_i.$$ Prove that $$\lim_{n\rightarrow\infty} \frac{M_n}{\sqrt{2\log n}}=1\quad\text{a.s.}$$ I used the fact that…
15
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Different versions of functional central limit theorem (aka Donsker theorem)?

I have seen several versions of functional central limit theorem (see the end of this post). I am confused, and hope someone could help to clarify their relations and differences. For example, I am wondering Do Billingsley's Probability and…
Tim
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If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

Let $f\in C^3(\mathbb R)$ be positive $g:=\ln f$ $d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^d)$ $\ell>0$, $\sigma_d:=\ell d^{-\alpha}$ for…
14
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4 answers

What is the probability that the sum of digits of a random $k$-digit number is $n$?

Let $X_1, X_2, \dots, X_k$ each be random digits. That is, they are independent random variables each uniformly distributed over the finite set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $S = X_1 + X_2 + \dots + X_k$. Given some large integer $n$, what…
ShreevatsaR
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13
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Intuitive explanation of Lyapunov condition for CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables: Lyapunov CLT. Let $s_n^2 = \sum_{k=1}^n \text{Var}[Y_i]$ and let $Y=\sum_i…
13
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2 answers

Limiting distribution of $\frac1n \sum_{k=1}^{n}|S_{k-1}|(X_k^2 - 1)$ where $X_k$ are i.i.d standard normal

Let $(X_n)$ be a sequence of i.i.d $\mathcal N(0,1)$ random variables. Define $S_0=0$ and $S_n=\sum_{k=1}^n X_k$ for $n\geq 1$. Find the limiting distribution of $$\frac1n \sum_{k=1}^{n}|S_{k-1}|(X_k^2 - 1)$$ This problem is from Shiryaev's…
12
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2 answers

Convergence in distribution of conditional expectations

I was just reading this question, which is about how the classical central limit theorem can be interpreted as giving a rate of convergence for the law of large numbers for iid random variables. I was wondering whether the same idea can be…
12
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1 answer

Limit theorems in measure theory

From probability theory/measure theory we know set of theorems such as Monotone convergence, dominated convergence or conditions like uniform integrability which deals with the general question of interchanging limits and integration. In these cases…
10
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4 answers

Delta epsilon proof with x approaches infinity

I have a rational function that reaches a horizontal asymptote as x approaches infinity. How would you do a delta-epsilon proof with $x\to\infty$. Here is the limit statement: $$\lim_{x\to\infty}\frac{3x+7}{2x-1} = \frac{3}{2}.$$ Hope some one can…
Paulo
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10
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$CLT$ and $LLN$ give different results

I tried to solve a problem two different ways and I got different results. Let $( X_i )_{i \in \mathbb{N}}$ be a series of independent, identically distributed random variables, with $\mathbb{E}[X_i] = 1$ and $\mathbb{V}[X_i] =…
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