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

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-3
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0answers
20 views

how to solve the current problem of evaluating limits approching zero [on hold]

$\sum{(\dfrac{1}{pi}\cdot \tau\cdot \alpha)X}, $$ \ \ \sum(\sqrt{(1-(\mho/2\alpha\tau) {k_a}^2)})$ first summation limit is $\tau=0$ second summation limit is $\mho=-2\alpha\tau$ to $+2 \alpha \tau$ ...
0
votes
3answers
33 views

Limit of a exponential sequence

I am stuck with this tricky limit... Any idea? $$ \lim \limits_{n \to \infty}(n^2e^{-1/n}+ne^{-1/n}-n^2) $$ Of course, I am not allowed to use l'Hopital...
0
votes
1answer
22 views

Limit with two parameters

Let: $$\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{{{n^2} + 2n + 1}}{{{{\left( {4{n^2} + 6n + 2} \right)}^{\frac{{2m}}{k}}}}}} \right)^{\frac{1}{{2m}}}}$$ if $m=2k$ it's easy to see that ...
1
vote
0answers
51 views

Donsker for randomly stopped processes

A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the ...
2
votes
3answers
31 views

Limit of a polynomic-exponential sequence

I have to calculate the following limit: $$L=\lim \limits_{n \to \infty} -(n-n^{n/(1+n)})$$ I get the indeterminate form $\infty - \infty$ and I don't know how to follow. Any idea? Thank you very ...
2
votes
0answers
13 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
0
votes
0answers
33 views

Limit of ratio of random sequences

Let $X_1, X_2, \dots $ i.i.d random variables with the following properties. (1) $\mathbb{P}(|X_j|>x)= x^{-\alpha}L(x)$, where $\alpha \in (0,1)$ and $L(x)$ is a slowly varying function. (2) ...
1
vote
1answer
35 views

A basic question on Lindeberg and Lyapunov condition

Suppose along with independence in each row of a triangular array, it is given that in each row random variables are identically distributed then what Lindeberg and Lyapunov condition reduces to ? ...
0
votes
1answer
24 views

How do I use limit laws to evaluate $\lim\limits_{ x \to \pi/2} [\tan(x) (\sin^2(x)-1)]$?

I'm having trouble with the limit laws.. especially when it comes to anything that has trig in it.
-3
votes
0answers
28 views

Motivation behind the expression of Lindeberg condition

What is the motivation behind the expression behind Lindeberg condition ? I am not understanding how he came up with that expression ? Once we assume that condition then it is immediate that for ...
0
votes
3answers
39 views

proving $\lim(1/y_n)=1/y$

I have a question on the proof of $\lim (1/y_n)=1/y$ under $\lim y_n=y$; $|y_n-y| < \epsilon |M|/|y|$ $y_n$ is convergent so that it's bounded by some number $|M|$; ...
2
votes
1answer
23 views

A Question about Limits

If I have a function $f(x) = g(x)/h(x)$, where $g$ and $h$ are polynomial functions, and I want: $ \lim_{x \to k} f(x)$, when $h(k) = 0$ Normally I would factor each one of the functions and cancel ...
1
vote
2answers
74 views

Using L'hopital's rule to solve problem.

Show that $$\lim_{x \to 0} \frac{-3x }{e^{x/3}}=0 $$ by L'hopital's rule. I know how to solve this without using L'hopital's rule. I was just reading about this and was wondering can we solve it ...
0
votes
1answer
26 views

limit of $f(x)g(x)$

If we know that $\lim_{x\to a}f(x)=\infty$, and $\lim_{x\to a}g(x)=0$, then what will be $\lim_{x\to a}(f(x)g(x))$? How do you prove the result? By using examples I can see that the limit can be ...
2
votes
2answers
60 views

Prove ${a_n} = \mathop {\lim }\limits_{n \to \infty } \sum\nolimits_{k \ge 1} {{1 \over {{k^2}}}} $ Converges by using Cauchy's criteria

Prove ${a_n} = \mathop {\lim }\limits_{n \to \infty } \sum\nolimits_{k \ge 1} {{1 \over {{k^2}}}} $ Converges by using Cauchy's criteria. What I did: Let $n, m=n+k \in \mathbb{N}$. $$\left| ...
2
votes
1answer
19 views

Partial limits and sets of indices

Let $\{a_n\}$, and $A_1...A_n \subseteq \mathbb{N}$, such that $A_1\cup A_2... \cup A_n= \mathbb{N}$. We denote $P_k$ as the set of all partial limits (= limit of a subsequence) with indices $\in$ ...
1
vote
1answer
23 views

Show there must be another “partial limit”

Let $P$ the set of all partial limits of the sequence $\{a_n\}$ (partial limit = limit of a subsequence). It's given that $\{0,2\} \subseteq P$ and $\forall n\in \mathbb{N}. \left| {a_{n+1}-a_n} ...
2
votes
1answer
31 views

Evaluating a limit with two steps - Right/Legal?

$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {\left( {{{4{n^2}} \over {(2n + 1)(2n - 1)}}} \right)^{1 - {n^2}}} = \mathop {\lim }\limits_{n \to \infty } {\left( {{1 \over {{{(2n + 1)(2n ...
0
votes
0answers
20 views

How to generalize the convergence together lemma

How can we generalize the convergence together lemma that says: If $X_n \to X_{\infty}$ in distribution and $Y_n \to c$ ; $c$ is a constant in distribution/probability . Then $ X_n+Y_n \to ...
0
votes
1answer
38 views

Proving that limit does not exist

If we define $$ f(x)=\begin{cases} x&x\geq0\\ -1&x<0 \end{cases} $$ To prove that $\lim_{x\to0}f(x)$ does not exist, what am I required to do? I already know that if $\lim_{x\to ...
0
votes
2answers
54 views

Use the squeezing theorem to find the limit of the sequence

Is anyone able to help me answer this question? Or point me in the right direction? Use the squeezing theorem to find the limit of the sequence $\{a_n\}_{n=1}^{\infty}$ with $n$-th term ...
0
votes
2answers
36 views

Showing $1$ is a limit of a sequence

Let $x_n$ be a sequence such that: $x_1 = {3 \over 2}$ and $x_n = {3 \over {4-x_{n-1}}}$. I already showed by induction that $x_n$ is strictly decreasing and bounded below by $1$. Is is suffice in ...
2
votes
2answers
47 views

Limit question involving L'Hospitals rule

I need help in solving the below question using L'Hospitals rule: $$ \lim_{x \rightarrow 0} \left( \frac{1}{x^2}-\cot^2 x \right) $$ I'm getting infinity as the final answer. Thanks in advance.
2
votes
1answer
36 views

Help with the squeeze theorem

Is anyone able to help me/point me in the right direction with this question? Use the squeezing theorem to find the limit of the sequence $\{a_n\}_{n=1}^{\infty}$ with $n$-th term ...
1
vote
1answer
24 views

Texbook typo? Divergence Criterion for Functional Limits

The last sentence should say "lim f(x) does not exist", rather than "lim f(c) does not exist", correct?
1
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0answers
26 views

Application of the Weak Law of Large Numbers.

I have in my problem that $X_1,\ldots,X_n$ is a random sample from a distribution with probability density $f(x; \theta)=\theta x^{\theta-1}, 0<x<1$. Furthermore, $-\log X_i ...
2
votes
2answers
66 views

Showing that Lindeberg condition does not hold

Let $X_1, X_2, \dots$ be independent random variables and $$X_n = Y_n + Z_n$$ where $Y_n$ takes values $1$ and $-1$ with chance $1/2$ each, and $$P(Z_n = \pm n) = 1/(2n^2) = (1 - P(Z_n = 0))/2$$ and ...
3
votes
3answers
74 views

Find limit of the following sequence using squeeze theorm

I'm looking for some help on these Q's. Find the limit of the following sequences if it exists: 1) $$ a_n = \sqrt[n]{2^n+4^n+5^n} $$ 2) $$a_n = \frac{3^n}{n+n!}; n!= 1*2...n $$ Can someone show ...
2
votes
2answers
42 views

Application of Central Limit Theorem for nonnegative RV

Suppse $X_1, X2, \dots$ are iid nonnegative r.v.s with mean 1 and finite variance $\sigma^2>0$. Show that $2(\sqrt{S_n}-\sqrt{n}) \rightarrow \mathcal{N}(0,\sigma^2)$
1
vote
1answer
38 views

Find the limiting distribution of the following random variable

Let $X_1,X_2,...$ Be independent random variables with common density: $$f_X(x)=\alpha x^{-(\alpha+1)}. x>1$$ Where $\alpha>0$. Define a new sequence of random variables: ...
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vote
4answers
50 views

a simple problem on limit made me confused

I tried to solve this problem as $$\lim_{x \to \frac{\pi}{2}} (1+\cos x)^{\tan x}$$ if $y=(1+\cos x)^{\tan x} \implies \log y=\tan x \log(1+\cos)$ $$\lim_{x \to \frac{\pi}{2}}\log y=\lim_{x \to ...
1
vote
1answer
47 views

Asymptotic Relative Efficiency: Poisson

I'm trying to find the asymptotic relative efficiency of a Poisson process: $$\frac{\lambda^t \exp(-\lambda)}{t!} = P(X=t).$$ When $X = t = 0$, the best unbiased estimator of $e^{-\lambda}$ is ...
0
votes
0answers
30 views

Compute imaginary part of the limit

I am trying to compute the imaginary part of this particular question $$\lim_{z\rightarrow i} \frac{iz^3}{z+9 i}$$
0
votes
0answers
20 views

Find $\alpha$ such that $\lim_{(x, y) \to (0, 0)} \frac{|x|^\alpha y}{x^2 + y^2} = 0$. Can I simply move to polar coordinates?

This is the full exercise: Given function: $$f(x, y) = \begin{cases} \frac{|x|^\alpha y}{x^2 + y^2}, & \text{if }(x,y) \ne (0, 0) \\ 0, & \text{if } (x,y) = (0,0) \\ \end{cases}$$ ...
0
votes
2answers
24 views

Is $\lim_{n\rightarrow\infty }nz^{n!n}=0$ for $|z|<1$?

Is $\lim_{n\rightarrow\infty }nz^{n!n}=0$ for $|z|<1$? We have a $\infty \cdot 0$ case, then how we proceed? How to use the L'Hospital's Rule? Thanks in advance!
1
vote
4answers
51 views

determine the limit of sequence?

Need help on how to determine the limits, if it exists for this sequence. Have no idea where to start. Thanks in advance smart people! $$A_n = \dfrac {(n+2)^{2n}}{(n^2-n-6)^n}$$
1
vote
2answers
61 views

how to find existence and value of limit in multivariable calculus

I was in maths class and i found a question interesting. Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar ...
0
votes
3answers
27 views

Simple Limit Question with Denom 0

I was asked to find the limit of the following: $$\lim_{x\to -1}\frac{\sqrt {x ^ 2 + 8} - 3}{x + 1} $$ I have tried using the Limit Laws but am always getting $ \frac {0}{0} $. The answer given is ...
2
votes
1answer
59 views

central limit theorem for high dimensional random walk

Consider random walk in $\mathbb{Z}^d$, $d>1$, with $x(t) = x(t-1) + \xi$, where $\xi$ has some probability distribution in $\mathbb{Z}^d$ with finite support, expectation $m = \sum_{v \in ...
1
vote
1answer
58 views

Problem related to Fatou's Lemma (Measure theory)

Statement of the problem: Show that if $\{f_n\}_{n=1}^{\infty}$ is a sequence of non-negative measurable functions on $\mathbb{R}^d$, then for any $t>0$ we have: $$m\left(\left\{x\in ...
0
votes
1answer
26 views

functions which satisfy the following limit

I need to find the functions, f(x), for which the following is satisfied (or alternatively the functions for which it is not): $$\text{lim}_{\:x\to \infty} \large{{f(x)\over e^{-x^m\over2}}} \to ...
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vote
0answers
28 views

limit formalisms

Let $f:\mathbb R \to \mathbb R$ be a function and $a\in \mathbb R$ a point. The Cauchy definition of the limit $\lim _{x\to a}f(x)=L$ is well-known. For pedagogical reasons I'm interesting in a ...
4
votes
1answer
41 views

Triangular arrays and almost sure convergence of row averages

Suppose we have the triangular array $\{\{X_{in},i=1,\ldots,n\},n=1,2,\ldots\}$: $$\begin{array}{ccccc} X_{11}&&&&\\ X_{12}&X_{22}&&&\\ ...
0
votes
0answers
35 views

Show gamma-function $\Gamma(r)$ is well-defined for any $r >0$ (the limit of the improper definite integral exists)

Introduction: I have proved the following: Suppose that Poisson events are occuring at the constant rate of $\lambda$ per unit time. Let the random variable $Y$ denote the waiting time for the rth ...
2
votes
1answer
79 views

A simpler solution to a limit question?

Okay I saw this limit question in Thomas' Calculus 12th Edition: $\lim \limits_{x \to 0} \frac{\tan3x}{\sin8x}$ The answer is $\frac{3}{8}$. I was able to get the correct answer using this ...
1
vote
1answer
72 views

Determine value of integral:$I=\int_0^1\frac{\ln(1+x)}{x}dx$

Determine value of integral:$$I=\int_0^1\frac{\ln(1+x)}{x}dx$$ I use Taylor's expansion with $x_0=0$, we have: $$\ln(1+x)=\sum_{i=1}^{\infty}\frac{(-1)^{i+1}x^i}{i}$$ Hence ...
1
vote
1answer
35 views

Application of the Dominated Convergence Theorem (probabilistic version).

I am currently working on the following problem and I think I've got the solution more or less, but there is a minor question about the usage of the Dominated Convergence Theorem. Let $f: [0,1] ...
1
vote
1answer
45 views

Dominating function for infinite fn

I am trying to find a dominating function $g(x)$ for the series of functions $\{f_n\}$ $$f_n = \frac{x~e^{-x/n}}{n^2}$$ The integral of the function $f_n$ I believe is a point mass of $1$ at $0$ when ...
0
votes
2answers
30 views

Convergence in Probability with a case environment

Suppose $\left(X_{j}\right)_{j\ \geq\ 1}$ is a sequence of independent random variables such that $$X_{j} = \begin{cases} j^{3}, &\text{with probability}\ \frac{1}{j}; \\[1mm] 1, &\text{with ...
1
vote
2answers
68 views

Need to justify $\int_{n}^{S_{n}}\frac{dx}{\sqrt{x} } \approx \frac{S_{n}-n}{\sqrt{n}}$

This is the text of the problem: Let $(X_{j})_{j\geq 1}$ be i.i.d. nonnegative with $E\{X_{1}\}=1$ and $\sigma_{X_{1}}^{2}=\sigma^{2} \in (0, \infty)$, and let $S_{n}=\sum_{j=1}^{n}X_{j}$. Show that ...