0
votes
1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
0
votes
0answers
61 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
-1
votes
0answers
12 views

Let $X_n(s)=o_p(1),\forall s$. Can we infer that $\|X_n\|=o_p(1)$? [closed]

Let $X_n(s)=o_p(1),\forall s$ where $X_n(s)$ is a sequence of random functions. Can we infer that $\|X_n\|=o_p(1)$ where $\|.\|$ is a norm of function $f(s)$?
1
vote
0answers
36 views

Sums of Power Law random variables

Suppose $F$ be a pareto distribution with scale parameter $x_m$ and shape parameter $\alpha$. Assume $X_1, X_2 , ..., X_n$ are iid random variables drawn from $F$. Let $S_n(k) = X_1 ^k + X_2 ^k + ...
0
votes
1answer
26 views

Sum of bounded in probability random variables

I'm self-studying probabilistic order notation, and I wanted to show some properties to get used to it. But now I'm having trouble showing that the sum of two random variables that are bounded in ...
3
votes
1answer
48 views

Prove $\lim_{n \to \infty} \frac{\Gamma(n+1/2)}{\Gamma(n)~n^{1/2}}=1$

Prove $$\lim_{x \to \infty} \frac{\Gamma(x+1/2)}{\Gamma(x)~x^{1/2}}=1.$$ I got this problem from Probability and Statistics by Degroot & Schervish. There is a hint to use Stirling's formula ...
1
vote
0answers
121 views

Limit of sequence of integral related i.i.d. observations

Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat ...
0
votes
1answer
41 views

Expected number of $k$-cliques in $G(n, 1/2) \ge 1$

Let the expected number of $k$-cliques be denoted by $$f(k) = \binom{n}{k} (\frac{1}{2})^{- \binom{k}{2}}$$ let $k_0$ denote the largest $k$ such that $f(k) \ge 1$. I want to prove that $k_0 = ...
1
vote
1answer
73 views

Order related to Empirical distribution function and Normal distribution

Let $X_1,\dots,X_n$ are i.i.d with distribution function $F$. Let $\hat F_n$ be its empirical distribution function, i.e., $$ \hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x) $$ where $1_A(x)$ is the ...
0
votes
0answers
34 views

Relations between stochastic Big O-Notation, limsup and lim?

I have to solve a list of exercises on probability theory but I'm having several problems in understanding the following questions; could you help me? Thank you! Let $||.||$ indicate the Euclidean ...
0
votes
1answer
57 views

$\frac{1}{n} \sum_{k=1}^{n-s} X_{k+s}X_{k}$ the same as $\frac{1}{n} \sum_{k=1}^{n} X_{k+s}X_{k}$ for $n \rightarrow \infty$?

I need to show that $$ \frac{1}{n} \sum_{k=1}^{n-s} X_{k+s}X_{k}$$ for some number $s$ is essentially the same (asymptotically negligible) as $$ \frac{1}{n} \sum_{k=1}^{n} X_{k+s}X_{k}$$ as $n ...
2
votes
1answer
76 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 ...
25
votes
3answers
496 views

Expected length of the shortest polygonal path connecting random points

$N$ points are selected in a uniformly distributed random way in a disk of a unit radius. Let $L(N)$ denote the expected length of the shortest polygonal path that visits each of the points at least ...
3
votes
0answers
88 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of ...
1
vote
2answers
121 views

When can we exchange expectation and maximum for asymptotic results?

Motivated in the analysis of algorithms, consider the following setup. Assume we have discrete random variables $X^{(n)}_1, \dots, X^{(n)}_n$ which we can not assume to be identical or independent. ...
1
vote
1answer
76 views

Question on Convergence in Probability

I appreciate if you could guide me on this question: Assumptions: $X_n \rightarrow^p c$: $X_n$ convrges in probability to a constant c. g(.) is any function that satisfies: $$\text{if } a_n - c = ...
12
votes
2answers
338 views

Asymptotics of the sum of squares of binomial coefficients

We are trying to estimate the cardinality $K(n,p)$ of so-called Kuratowski monoid with $p$ positive and $n$ negative linearly ordered idempotent generators. In particular, we are interesting in the ...
2
votes
1answer
100 views

Large Deviations Problem

Let $\left(X_n\right)_{n\geq 1}$ be i.i.d random variables on $\left(\Omega,\mathcal A, \mathbb P\right)$, $X_1$ with mean $\mu$, and $$ L(\lambda) = \begin{cases} \log\mathbb E\left(e^{\lambda ...
11
votes
1answer
182 views

Calculate Asymptotics of Integral?

Let $f$ be a continuous function on $[0,1]$. How do I calculate the asymptotics, as $n\rightarrow\infty$, of $\displaystyle \int_{[0,1]^n}f\left(\frac{x_1+...+x_n}{n}\right)\text d x_1...\text d ...
0
votes
1answer
161 views

Asymptotic Notation more specifically, Big-O notation

How the functions in the class $O(d)^d$ and $\epsilon^{1/O(d.4^d)}$ looks like..? where $\epsilon$<1. I am really confused with this complicated Big-O notations Can you please help me out.
3
votes
2answers
278 views

$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that ...
2
votes
2answers
192 views

Help with the integral for the variance of the sample median of Laplace r.v.

When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.: ...
0
votes
0answers
43 views

understanding of 1-unconditionality

Let $X=(X, \|\cdot\|_X)$ be normed space with $x_1, \ldots, x_m\in X$. Assume, $\int_{[-1,1]^m}\|\sum_{i=1}^ma_ix_i\|_Xd\mu(a)=1$, where $\mu$ is the Lebesgue measure on $[-1,1]^m$, $a \in [-1,1]^m$. ...
3
votes
0answers
143 views

How to solve equation involving binomial coefficient?

I'm reading this paper which says If we have $$ \binom n d p^{\binom d 2} = 1 $$ where $ 0 < p \le 1$, then $$ d = 2 \log_bn - 2 \log_b \log_b n + 2 \log_b\left(\frac 1 2 e\right) + 1 + O(1) ...
1
vote
1answer
417 views

Observed information matrix is a consistent estimator of the expected information matrix?

I am trying to prove that the observed information matrix evaluated at the weakly consistent maximum likelihood estimator (MLE), is a weakly consistent estimator of the expected information matrix. ...
4
votes
2answers
2k views

Central Limit Theorem and sum of squared random variables

This is a two-part question. Suppose I am drawing random variables $X_i\sim A$, $1\leq i \leq n$ where $A$ is a zero-mean, finite variance $\sigma_A^2$, symmetric probability distribution having ...
0
votes
2answers
160 views

Asymptotic probability: boys and girls in a line

We have $n$ people: $\alpha n$ are boys and $(1-\alpha)n$ are girls. They are standing in a line in a random order. We pick up one boy also at random. What can one say about the probability that ...