Tagged Questions

0
votes
2answers
25 views

Central value of the partial exponential function [duplicate]

I need help calculating the central value of the partial exponential function : $$\lim_{n \to \infty} e^{-n} \sum^n_{k=0} \frac{n^k}{k!}$$ fd
1
vote
1answer
31 views

Limit involving probability

Let $\mu$ be any probability measure on the interval $]0,\infty[$. I think the following limit holds, but I don't manage to prove it: $$\frac{1}{\alpha}\log\biggl(\int_0^\infty\! x^\alpha ...
3
votes
1answer
132 views

How to bound the maximal consecutive length in a random subset of [n] as function of n?

Let $S$ be a random subset of $[n]=\{1,2,\ldots,n\}$ chosen uniformly from $[n]$'s subsets. How can I find a function $f(n)$ s.t. for any $\varepsilon \gt 0$, $$\lim_{n \rightarrow \infty} P\left[(1- ...
3
votes
2answers
50 views

Independence is preserved under taking almost sure limits

I´m not sure why this theorem is right, how can i prove it? Let $X_1,X_2.... $ and $Y_1,Y_2...$ be random variables such that $X_n,Y_n$ are independent for every $n∈\mathbb N$ and such that X, resp ...
2
votes
1answer
91 views

Set of measure zero?

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W)=1$. Consider a measurable function $f:W \rightarrow \mathbb{R}_{\geq 0}$. Say if the following holds true. $$ \lim_{M ...
2
votes
1answer
67 views

How can I show $\lim_{n\to\infty}\prod_{j}^n\frac{\sin(jn^{-3/2}u)}{jn^{-3/2}u}=e^{-u^2/18}$?

This is from and exercise of probability theory: Let $(X_j)_{j\geq 1}$ be independent and let $X_j$ have the uniform distribution on $(-j,j)$. Show that $$ ...
2
votes
2answers
66 views

What is the limit of a sequence of events? Probability

Here's a question I'm sturglling with: Show that for an increasing sequence of events $$A_1\subset A_2\subset A_3\subset ...$$ the next equation holds ...
0
votes
1answer
203 views

Limit of sum of indicator function

I came across a problem involving the following limit: $\lim_{n \to \infty} (\frac{1}{n} \sum\limits_{i=1}^n \mathbf{1}_{x_i>0}), \mbox{ where } X \sim N(\mu, \sigma)$ How would you approach ...
5
votes
2answers
116 views

Convergence to the stable law

I am reading the book Kolmogorov A.N., Gnedenko B.V. Limit distributions for sums of independent random variables. From the general theory there it is known that if $X_i$ are symmetric i.i.d r.v ...
2
votes
1answer
96 views

Exercise: Limits and Probability Measure

Let $\mu$ be a probability measure on $X$ (closed but unbounded), so that $\int_X \mu(dx) = 1$. Let functions $f_i:X \rightarrow \mathbb{R}_{\geq 0}$, $i = 1,2,...$, be Uniformly Integrable. Prove ...
1
vote
1answer
72 views

Sequences in a Probability Measure…

Let $\mu$ be a probability measure over the (closed but unbounded) set $X \subseteq \mathbb{R}^m$: $\int_X \mu(dx) = 1$. Consider function $f:\mathbb{R}^n \times \mathbb{R}^n \times X \rightarrow ...
3
votes
2answers
1k views

Transform uniform distribution to normal distribution using Lindeberg–Lévy CLT

Currently i am developing a game which is based on many computations of random values and therefore i have implemented many algorithms like the Mersenne-Twister etc. Unfortunately, all generators ...
2
votes
1answer
378 views

Fatou's Lemma and Almost Sure Convergence (Pt. 1)

I have a question regarding Fatou's Lemma and a sequence of random variables converging almost surely. Fatou's Lemma states If $\forall n \in \mathbb{N}, \,\, X_{n} \ge 0$ and $\displaystyle X = ...
8
votes
2answers
229 views

Limits of Expectations

I've been fighting with this homework problem for a while now, and I can't quite see the light. The problem is as follows, Assume random variable $X \ge 0$, but do NOT assume that ...