Uniformly Distributed Sequence

In summary, the given sequence is defined as a_j = j/n for n integers. We are asked to prove that as n approaches infinity, the proportion of integers in the set [a,b] approaches b-a. The proof involves using combinatorics and the law of large numbers.
  • #1
snipez90
1,101
5
Let [tex]\left(a_n\right)[/tex] be the sequence

[tex]\frac{1}{2},\, \frac{1}{3},\, \frac{2}{3},\, \frac{1}{4},\, \frac{2}{4},\, \frac{3}{4},\, \frac{1}{5},\, \frac{2}{5},\, \frac{3}{5},\, \frac{4}{5},\, \frac{1}{6},\, \frac{2}{6},\,\mbox{...}[/tex]

Suppose that [tex]0\leq a<b \leq 1.[/tex] Let [tex]N(n;a,b)[/tex] be the number of integers [tex]j \leq n[/tex] such that [tex]a_j \in \left[a,b\right].[/tex] Prove that
[tex]\lim_{n\rightarrow \infty}\frac{N(n;a,b)}{n} = b-a.[/tex]

I already know how to do this based on the definition of a sequence. The basic idea is to take the set of rational numbers {1/n, 2/n, ... , (n-1)/n} for an arbitrary n and consider the smallest member of the set which is also in [a,b], giving us a bound on a, and a similar consideration for the largest member of the set in [a,b] gives a bound on b. This allows us to estimate the number of elements of the set (for that particular n) that are also in [a,b].

My proof of this was rather long, but entirely elementary (the source is Spivak). I was wondering if there are more sophisticated methods of dealing with this type of problem. Thanks in advance.
 
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  • #2
It is combinatorics by nature ending with the law of large numbers. Any proof will have to bridge this gap, so the answer is probably not, although it is hard to tell without knowing your proof.
 

1. What is a Uniformly Distributed Sequence?

A Uniformly Distributed Sequence is a type of sequence in mathematics and computer science where the values occur with equal probability, meaning there is no bias towards any specific value. In other words, each value in the sequence is equally likely to occur.

2. How is a Uniformly Distributed Sequence different from a Random Sequence?

A Uniformly Distributed Sequence is a specific type of sequence where the values occur with equal probability, whereas a Random Sequence can have any distribution of values, including non-uniform distributions. In a Uniformly Distributed Sequence, each value is equally likely to occur, while in a Random Sequence, some values may occur more frequently than others.

3. Why is a Uniformly Distributed Sequence important?

A Uniformly Distributed Sequence is important because it is often used in simulations and statistical analysis. It allows for a fair representation of all possible outcomes and avoids any bias towards specific values. This makes it a useful tool for modeling real-world scenarios and making accurate predictions.

4. How is a Uniformly Distributed Sequence generated?

A Uniformly Distributed Sequence can be generated using various methods, depending on the specific application. In computer science, it can be generated using algorithms such as the linear congruential generator or the Mersenne Twister. In mathematics, it can be generated using mathematical formulas or physical processes like rolling dice.

5. Can a Uniformly Distributed Sequence be infinite?

Technically, a Uniformly Distributed Sequence can be infinite, but in practice, it is limited by the capabilities of the computer or the precision of the measuring instrument. In many cases, a large enough sample of values can be considered as effectively infinite for practical purposes. However, in some applications, such as theoretical mathematics, an infinite sequence may be necessary to prove certain properties or theorems.

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