Steven R. Dunbar
Department of Mathematics
203 Avery Hall
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Topics in
Probability Theory and Stochastic Processes
Steven R. Dunbar

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Asymptotic Limits

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_______________________________________________________________________________________________ ### Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

_______________________________________________________________________________________________ ### Section Starter Question

Name some sequences ${s}_{n}$ and ${t}_{n}$ such that $lim{s}_{n}=\infty$ and $lim{t}_{n}=\infty$ and

$\underset{n\to \infty }{lim}\frac{{s}_{n}}{{t}_{n}}=1.$

What does this say about the rate at which ${s}_{n}$ and ${t}_{n}$ approach $\infty$ ?

_______________________________________________________________________________________________ ### Key Concepts

1. For sequences ${s}_{n}$ and ${t}_{n}$, we say that ${s}_{n}$ is asymptotic to ${t}_{n}$, written ${s}_{n}\sim {t}_{n}$, if $\underset{n\to \infty }{lim}\frac{{s}_{n}}{{t}_{n}}=1$.

__________________________________________________________________________ ### Vocabulary

1. For sequences ${s}_{n}$ and ${t}_{n}$, we say that ${s}_{n}$ is asymptotic to ${t}_{n}$, written ${s}_{n}\sim {t}_{n}$, if $\underset{n\to \infty }{lim}\frac{{s}_{n}}{{t}_{n}}=1$.

__________________________________________________________________________ ### Mathematical Ideas

Definition. For sequences ${s}_{n}$ and ${t}_{n}$, we say that ${s}_{n}$ is asymptotic to ${t}_{n}$, written ${s}_{n}\sim {t}_{n}$, if $\underset{n\to \infty }{lim}\frac{{s}_{n}}{{t}_{n}}=1$.

Lemma 1 (Reflexivity). If ${s}_{n}\sim {t}_{n}$ then ${t}_{n}\sim {s}_{n}$.

Lemma 2 (Transitivity). If ${s}_{n}\sim {t}_{n}$ and ${t}_{n}\sim {u}_{n}$, then ${s}_{n}\sim {u}_{n}$.

Lemma 3 (Multiplication I). If ${s}_{n}\sim {t}_{n}$ and ${u}_{n}\sim {v}_{n}$ as $n\to \infty$, then ${s}_{n}{u}_{n}\sim {t}_{n}{v}_{n}$.

Lemma 4 (Zero Asymptotic Limits). If ${s}_{n}\sim {t}_{n}$ and ${s}_{n}\to 0$, then ${t}_{n}\to 0$ as $n\to \infty$.

Lemma 5 (Logarithms). If ${s}_{n},{t}_{n}>0$, and ${s}_{n}\sim {t}_{n}$ and $\underset{n\to \infty }{lim}{s}_{n}=0$, then $ln\left({s}_{n}\right)\sim ln\left({t}_{n}\right)$.

Proof. Let $\eta >0$ be given. Since ${s}_{n}\sim {t}_{n}$, and $lim{s}_{n}=0$, then too $lim{t}_{n}=0$. Hence $limln\left({t}_{n}\right)=-\infty$. Choose ${N}_{1}$ so large that

$max\left(ln\left(9∕10\right)∕ln\left({t}_{n}\right),-ln\left(11∕10\right)∕ln\left({t}_{n}\right)\right)<\eta$

for $n>{N}_{1}$. Choose ${N}_{2}$ so large that

${t}_{n}\cdot \left(9∕10\right)<{s}_{n}<{t}_{n}\cdot \left(11∕10\right).$

Hence

$ln\left({t}_{n}\right)+ln\left(9∕10\right)

Then

$1+\frac{ln\left(9∕10\right)}{ln\left({t}_{n}\right)}>\frac{ln\left({s}_{n}\right)}{ln\left({t}_{n}\right)}>1+\frac{ln\left(11∕10\right)}{ln\left({t}_{n}\right)}.$

For $n>max\left({N}_{1},{N}_{2}\right)$

$1-\eta <\frac{ln\left({s}_{n}\right)}{ln\left({t}_{n}\right)}<1+\eta$

so $ln\left({s}_{n}\right)\sim ln\left({t}_{n}\right)$. □

Lemma 6 (Multiplication II). If ${s}_{n}\cdot {u}_{n}\sim {t}_{n}$ and ${u}_{n}\ne 0$, then ${s}_{n}\sim \frac{{t}_{n}}{{u}_{n}}$.

Lemma 7 (Substitution). If ${s}_{n}\sim {u}_{n}\cdot {t}_{n}$ and ${u}_{n}\sim {v}_{n}$, then ${s}_{n}\sim {v}_{n}\cdot {t}_{n}$.

Proof. If ${s}_{n}\sim {u}_{n}\cdot {t}_{n}$, and ${u}_{n}\sim {v}_{n}$ then $lim\frac{{s}_{n}}{{u}_{n}\cdot {t}_{n}}=1$ and $lim\frac{{u}_{n}}{{v}_{n}}=1$. Then multiplying these two limits together, $lim\frac{{s}_{n}}{{v}_{n}\cdot {t}_{n}}$ so ${s}_{n}\sim {v}_{n}\cdot {t}_{n}$. □

#### Sources

This section is just simple analysis of limits organized into elementary lemmas for later reference.

_______________________________________________________________________________________________ ### Problems to Work for Understanding

1. Let $0. Let ${k}_{n}=⌈np⌉$ be the least integer greater than or equal to $np$. Then show that ${k}_{n}\sim np$.
2. Prove the Reflexivity and Transitivity Lemmas and show that the “asymptotic to” relation among sequences is an equivalence relation.
3. Prove the Multiplication I and Multiplication II Lemmas

__________________________________________________________________________ ### References

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