Steven R. Dunbar

Department of Mathematics

203 Avery Hall

University of Nebraska-Lincoln

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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Mathematically Mature: may contain mathematics beyond calculus with proofs.

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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 $ ?

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

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

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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 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

$$max\left(ln\left(9\u221510\right)\u2215ln\left({t}_{n}\right),-ln\left(11\u221510\right)\u2215ln\left({t}_{n}\right)\right)<\eta $$

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

$${t}_{n}\cdot \left(9\u221510\right)<{s}_{n}<{t}_{n}\cdot \left(11\u221510\right).$$

Hence

$$ln\left({t}_{n}\right)+ln\left(9\u221510\right)<ln\left({s}_{n}\right)<ln\left({t}_{n}\right)+ln\left(11\u221510\right).$$

Then

$$1+\frac{ln\left(9\u221510\right)}{ln\left({t}_{n}\right)}>\frac{ln\left({s}_{n}\right)}{ln\left({t}_{n}\right)}>1+\frac{ln\left(11\u221510\right)}{ln\left({t}_{n}\right)}.$$

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 ${s}_{n}\sim {v}_{n}\cdot {t}_{n}$. □

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

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

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Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1

Email to Steve Dunbar, sdunbar1 at unl dot edu

Last modified: Processed from LATEX source on October 3, 2011