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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.

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2answers
18 views

Convergence in probability towards a constant

Prove that if a sequence $X_1,X_2..$ of random variables satisfies the following conditions: $$1.lim_{n\to\infty}\mathbb{E}(X_n)=a,\space a\in\mathbb{R}$$ $$2.lim_{n\to\infty}\mathbb{V}(X_n)=0$$ Then ...
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1answer
20 views

Higher Dimensional Random Walks

I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance ...
1
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1answer
23 views

Proving Brownian motions have bounded quadratic variation

I am looking into arguments which yield the result $(dB_t)^2=dt$ and authors first start by letting P be a partition $P=\{t_0, t_1, ..., t_n\}$ of the interval $ [0,T]$ where $t_i=\frac{i}{n}T$ and ...
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4answers
280 views

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 + ...
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0answers
112 views

Where does the bias come from in LogLog and HyperLoglog?

In the original LogLog paper from P. Flajolet, the cardinality estimation function is presented as $$E := \alpha_m m2^{1 \over m} {^{\sum_{j} M^{(j)}}}$$ Where $\alpha$ is needed to correct the ...
1
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1answer
30 views

Empirical distribution of sorted Gaussian numbers

I wrote a small program that does the following : Pick $N$ independent standard Gaussian numbers (expected value : 0, standard deviation : 1). Call that list $L=\{y_1, \ldots, y_N\}$. Sort that list ...
8
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1answer
180 views

$X_1,..,X_n$ i.i.d random variable and discrete, local limit theorem

Let $X_1,..., X_n$ be i.i.d discrete random variables which take their value in $\mathbb{Z}$ with a non trivial and finite support. Let $S_n = X_1+...+ X_n$ Prove the existence of $ 0 < C_1 &...
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0answers
25 views

CLT for decaying random variables

Suppose $X_1,X_2,...$ are bounded random variables with compact support, and $\frac{X_1+...+X_n}{\sqrt{n}}\overset{d}{\longrightarrow}N(0,1)$. Is there neccessarily a central limit theorem for the ...
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1answer
41 views

How to prove that the following estimators are biased and consistent?

Given a random variable $X$ following a geometric distribution with parameter $p.$ Then one estimator that can be obtained by considering the second moment $E[X^{2}]=\frac{2-p}{p^2}$, which is $$\hat{...
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2answers
33 views

$\lim_{x\rightarrow+\infty} x^{\alpha+\beta}P(|X|>x)=0\Leftrightarrow\lim_{x\rightarrow+\infty}x^{\alpha}E||X|^{\beta}\cdot1_{\{|X|>x\}}|=0$

I have met with a probability problem which I have no idea to deal with. It says: Let $\alpha>0$, $\beta\geq0$, prove: $\lim_{x\rightarrow+\infty} x^{\alpha+\beta}\mathbb{P}(|X|>x)=0\...
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0answers
7 views

Convergence of row superposition of null array to Poisson point process

Background: The limit of superposition of infinite number of independent point processes is a Poisson process under certain conditions. The conditions for the limit process to exist and be Poisson is ...
3
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3answers
125 views

Limiting a sequence of moment generating functions

I was trying to solve the following problem: Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables with the probability mass function $P\{X_n = \pm1 \} = \frac{1}{2}$, $n \in \...
1
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2answers
36 views

Probability limits of random variable sums

I have $X_1, X_2, X_3, \cdots$ which are independent random variables with the same non-zero mean ($\mu\ne0$) and same variance $\sigma^2$. I would like to compute $$\lim_{n\to\infty} P[\frac{1}n\sum^...
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1answer
12 views

Does this converge in distribution in this case?

Let $X_i$ be i.i.d. sequence of random variables with finite second moment. Then, by the Central limit theorem, $\frac{\sqrt{n}(\bar{X}-E(X))}{\sigma_X} \to N(0,1)$ in distribution. Let $\alpha_n$ ...
1
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1answer
52 views

Show that $n \mathbb{P}\{|X_1| \geq \epsilon \sqrt{n}\} \to 0$

Let $(X_n)_n$ be a sequence of identically distributed random variables with $\mathbb{E}X_1^2 < \infty$. Show that $$\lim_{n \to \infty} n\mathbb{P}\{|X_1| \geq \epsilon \sqrt{n}\} = 0$$ for all ...
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1answer
30 views

What is the rate of growth of $M_n := \max_{1 \le i \le n} U_i^{(n)}$, where $U_i^{(n)} \sim \operatorname{Uniform}[0,n]$?

On pp. 370-374 (see this previous question) of Cramer's 1946 Mathematical Methods of Statistics, the author shows that for any continuous distribution $P$, if $X_1, \dots, X_n \sim P$ are i.i.d., then ...
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1answer
22 views

Does weak convergence imply time average convergence?

I was reading some lecture notes for Poisson processes, in which after proving the following, $\lim_{t\rightarrow \infty}P(X(t)=j) = e^{-\rho}\frac{\rho^j}{j!}$ it states that this implies ...
4
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1answer
79 views

Let $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$ where $\beta\in(0,1)$, then $S_n/n^{(3-\beta)/2)}\Rightarrow c\chi$

Suppose $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$, where $\beta>0$. Show that: (i) If $\beta>1$ then $S_n\to S_\infty$ a.s. (ii) If $\beta\in(0,1)$ then $S_n/n^{(3-...
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2answers
79 views

A probability inequality: probability that the normalized sum of i.i.d. random variables is bounded below, is bounded below

I have been told that the following fact is true. Let $X_1,X_2,X_3,\dots$ be i.i.d. random variables. Then there exists $\epsilon$ such that for all $n$, $$\mathbb{P}\left(\frac{|X_1+\dots+X_n|}{\sqrt{...
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1answer
31 views

Find the limiting probability…

I have the following problem: Two men are shooting at a target. We can call them $X$ and $Y$. $X$ shoots after each hit and $Y$ after each miss. Their respective probabilities of hitting the target ...
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0answers
19 views

Showing that inequality holds with high probabilty

$X=\sum_{i=1}^{n^2} X_{i}$ random variable. $E[X]=\frac{n^2}{a(n)}$. For which $a$, $X \geq n\cdot C$ holds with high probability? So the following: $P(X\geq n\cdot C)= 1-o(1)$. What would be an ...
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0answers
24 views

estimating the sum of Cauchy variables by its biggest terms

Let $(X_1,...,X_n,...)$ be Cauchy variables. Let $(Y_{1,n},...Y_{n,n})$ be the same variables as $(X_1,...X_n)$, ordered by decreasing absolute value. The sum $\sum_{i=1}^n Y_{i,n}/n $ converges in ...
0
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1answer
29 views

$L^2$-quadratic variation of $X_t = B_t-tB_1$ for Brownian motion $(B_t)_{t \geq 0}$

I have a process $X_t = B_t - tB_1$, $B$ is a BM, $t\in[0,1]$. I have calculated its quadratic variation to be $t$, i.e.: $$\lim_{n\to\infty}V_n^t=\lim_{n\to\infty}\sum_{t_i\in E_n, t_{i+1}\leq t}(B_{...
2
votes
3answers
59 views

Calculate limit of \Gamma function

Show that $$\lim _{x \to \infty} \log \left( \frac{ \sqrt{x} \Gamma\left(\frac{x}{2}\right) } {\Gamma \left( \frac{x+1}{2}\right)} \right) = \frac{1}{2} \log(2),$$ where $\Gamma$ is the Gamma ...
2
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1answer
33 views

Suppose $X_n \rightarrow X$ a.s. and for each $n$, $X_n \perp \textit F$. Then is it true that $X \perp \textit F$?

For random variables $X_n$ and $X$, suppose $X_n \rightarrow X$ a.s. and for each $n$, $X_n \perp \mathcal F$ i.e. independent with $\mathcal F$. My question is: Is it true that $X \perp \mathcal ...
3
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0answers
37 views

Chebychev inequailty on random variable $X_1 + \ldots + X_n$

Let $X_i$ be independent random variable such that each $X_i$ is symmetric about 0 and Var($X_i) = 2i -1$, for $i \geq 1$. Then $$\lim_{n \mapsto \infty} P(X_1 + \ldots + X_n > n {\rm log}\;n) = ?$$...
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0answers
21 views

Central Limit Theorem and Continuity Theorem

Suppose $(X_n)_n$ is a sequence of r.v's with p.d.f's $|x|^{-3}$ outside of $(-1,1)$. I am trying to show that $Y:=\frac{\sum_i^n X_i}{\sqrt{2n\log n}}$ standard normal r.v. Solution: It is easy to ...
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0answers
15 views

probability density of a probability density of a probability density … limiting distribution?

probability density of a probability density of a probability density ... limiting distribution? If we start with a truncated $p(x)=N(0,1)$, with, say cutoffs at +/- 5, and compute the density $d_1(...
5
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1answer
127 views

Law of Iterated Logarithm when the mean does not exist

Let $X_1,X_2,\ldots$ be an i.i.d. sequence of random variables such that $X_1\geq 0$ a.s. and $\mathbb P[X_1>x]\sim x^{-\alpha}$, where $\alpha<1$. This implies that $X_1$ does not have finite ...
0
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1answer
33 views

Sequence of random variables that converges in distribution but whose variances does not converge to limit's variance

I am self-studying the Lehmann's amazing book "Elements of Large Sample Theory". There is a problem that I cannot figure out: 3.7 (page 122): Give an example in which $k_n(Y_n-c)$ tends in law to ...
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0answers
23 views

Convergence Indentity

Suppose $(X_n)_{n=1}^\infty$ is a sequence of independent r.v's such that $P(X_n =1)=P(X_n =-1)=1/2$. I am trying to show that $\sqrt{\frac{3}{n^3}}\sum_{i=1}^niX_i$ converges in distribution to the ...
2
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0answers
17 views

Estimate median of Cauchy distribution

Motivated by this question, assume we have independent samples $(X_i)_{i=1}^{\infty}$ from a Cauchy distribution with unknown median $a\in\mathbb{R}$ and scale parameter $b$. What is the best way to ...
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0answers
29 views

Population moment condition versus sample moment condition

Assume that the fourth moment of a random variable $X$ defined in $\left(\Omega, \mathcal{A}, P \right)$ satisfies: $$\mathbb{E} \left( X^4 \right) \leq C \left(\mathbb{E}(X^2) \right)^2, $$ for ...
2
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1answer
35 views

Is the existence of a finite variance enough for a CLT?

Suppose $X_1,X_2,...$ are random variables (not neccessarily i.i.d.) with $\mathbb{E}[X_i]=0$ such that $\operatorname{Var}\left(\frac{X_1+...+X_n}{\sqrt{n}}\right)\underset{n \longrightarrow \infty}{\...
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0answers
21 views

Rademacher Complexity Result

I was looking at one of the Rademacher Generalisation bound proofs, which says: If $G$ is a family of functions mapping from $Z$ to $[0, 1]$ and $\mathcal{R_m}(G)$ denotes the Rademacher Complexity ...
2
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1answer
37 views

Sum of Random Variables with no CLT

Suppose $Y_n$ are i.i.d. random variables with $E[Y_i]=0$, $\operatorname{Var}(Y_i)=C>0$, and denote $X_n=\frac{1}{n}Y_n$. Does $\sum_{i=1}^n X_i$ converge to $N(0,\sigma^2)$ ($\sigma^2=\frac{\pi}...
3
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1answer
88 views

Borel-Cantelli and “infinitely often”

The problem: Let $(X_n)_{n\geq 1}$ be a real-valued sequence of i.i.d. random variables and let $c > 0$. Use Borel-Cantelli's lemma to show that $$\sum_{n=1}^\infty P(X_n^2 > n) < \infty \...
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1answer
25 views

A question about a.s. convergence

While reading on almost sure convergence, I came across this equivalence: $$\{\omega \in \Omega: \lim_n X_n(\omega) = X(\omega)\} \equiv \bigcap\limits_{k=1}^{\infty}\bigcup\limits_{N=1}^{\infty} \...
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0answers
22 views

On Applying the Strong Law of Large Numbers in a prediction model accuracy metric

During the last few weeks I've been having a light discussion with some peers at work about the applicability of the Strong Law of Large Numbers (SLLN) to a certain batch of data. Everybody mantains ...
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1answer
54 views

Convergence of average of i.i.d. Bernoulli to Normal will be slower for p closer to 0 or 1

this is true. If you fit a normal to the data you see, if you have small samples, the normal is spread out in shape. So the area cut out by the $x=0$ line or $x=1$ line is larger than the case when $p=...
1
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1answer
30 views

Limit in distribution of a process bounded by another process.

Suppose that I have $\{Y_1^N\}_N$ and $\{Y_2^N\}_N$ two sequences of random variables defined in the same probability space $(\Omega, F, P)$. Suppose that $P(Y_1^N\leq Y_2^N)=1$ and that $Y_1^N\...
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1answer
79 views

Limiting distribution of $\frac{X_1+···+X_n}{\sqrt{X_1^2+···+X_n^2}}$ for $(X_i)$ i.i.d. logistic with mean $0$ and variance $2$

$X_1$, $X_2$, . . . are iid logistic random variables with mean $0$ and variance $2$, and $$T_n=\frac{X_1+···+X_n}{\sqrt{X_1^2+···+X_n^2}}$$Consider the sequence $T_1$, $T_2$, . . . and give the ...
2
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1answer
120 views

Probability a knight on a chess board is back where it started after $n$ moves

I'm trying to work my way through a problem concerning a random walk by a knight on a chessboard. I've modeled the board as a graph with 64 vertices and the random walk on the graph as a Markov chain ...
1
vote
1answer
167 views

Limiting Distribution of $\left(\prod\limits_{i=1}^{n} U_i\right)^{1/n}$ with $(U_i)$ i.i.d. uniform $(0,\theta)$

Let $(U_i)$ i.i.d. uniform $(0,\theta)$ and $$T_n=\left(\prod_{i=1}^{n} U_i\right)^{1/n}$$ Compute the limiting distribution of the sequence $(T_n)$. My try: $$ F_{T_n}(t) =\mathsf P(T_n \leq ...
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0answers
26 views

Relation between the biggest element and the second biggest element of Frechet random variables?

I am struggeling with the following: Imagine $X_1,...,X_n$ are iid. Frechet distributed random variables and imagine I order them by size so: $$X^{(1)}\leq \ldots \leq X^{(n)}$$ Now I am interested ...
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0answers
14 views

What is the limit of Frechet×Lognormal/Frechet?

Assume I have some Frechet r.v. $X$ distributed on the positive real numbers (loc=0, shape and scale some variables) and I multiply it with a log normal random variable $Y$ ($\mu=0$, $\sigma>0$) ...
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0answers
68 views

Convergence of expectations of a uniformly bounded process

Let $(\Omega, \mathcal{F}, (\mathcal{F}_{t})_{t \ge 0})$ be a filtered probability space. On it I have defined an $\mathcal{F}_{t}$-adapted process $(Y_{t})_{t \ge 0}$. It is a a continuous parameter,...
0
votes
1answer
40 views

Limit as $n \to \infty$ for Kelly Criterion

Suppose that with each game you have probability .7 of winning and .3 of losing. Say you bet half your money in each game. You start with $Y_0$ = 1. Let $Y_n$ be your bankroll after $n$ games and $$ ...
1
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0answers
28 views

Law of large numbers with continuous dependency

Suppose we have an i.i.d. sequence of random variables $X_n \sim X$, and a sequence $Y_n \to y_0$ in probability, where $y_0$ is a constant. Suppose also that $f : \mathbb{R}^2 \to \mathbb{R}$ is ...
1
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0answers
25 views

Convergence in probability of the multinomial sample correlation coefficient

This problem is from a Ph.D Qualifying Exam on mathematical statistics(also related to probability theory). Let $(X_1,\cdots,X_k)$ be a random vector with multinomial distribution of $n$ trials and ...