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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Rating

Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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Section Starter Question

Section Starter Question

Name some sequences sn and tn such that lim sn = and lim tn = and

lim nsn tn = 1.

What does this say about the rate at which sn and tn approach ?

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Key Concepts

Key Concepts

  1. For sequences sn and tn, we say that sn is asymptotic to tn, written sn tn, if lim nsn tn = 1.

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Vocabulary

Vocabulary

  1. For sequences sn and tn, we say that sn is asymptotic to tn, written sn tn, if lim nsn tn = 1.

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Mathematical Ideas

Mathematical Ideas

Definition. For sequences sn and tn, we say that sn is asymptotic to tn, written sn tn, if lim nsn tn = 1.

Lemma 1 (Reflexivity). If sn tn then tn sn.

Lemma 2 (Transitivity). If sn tn and tn un, then sn un.

Lemma 3 (Multiplication I). If sn tn and un vn as n , then snun tnvn.

Lemma 4 (Zero Asymptotic Limits). If sn tn and sn 0, then tn 0 as n .

Lemma 5 (Logarithms). If sn,tn > 0, and sn tn and lim nsn = 0, then ln(sn) ln(tn).

Proof. Let η > 0 be given. Since sn tn, and lim sn = 0, then too lim tn = 0. Hence lim ln(tn) = . Choose N1 so large that

max(ln(910) ln(tn), ln(1110) ln(tn)) < η

for n > N1. Choose N2 so large that

tn (910) < sn < tn (1110).

Hence

ln(tn) + ln(910) < ln(sn) < ln(tn) + ln(1110).

Then

1 + ln(910) ln(tn) > ln(sn) ln(tn) > 1 + ln(1110) ln(tn) .

For n > max(N1,N2)

1 η < ln(sn) ln(tn) < 1 + η

so ln(sn) ln(tn). □

Lemma 6 (Multiplication II). If sn un tn and un0, then sn tn un.

Lemma 7 (Substitution). If sn un tn and un vn, then sn vn tn.

Proof. If sn un tn, and un vn then lim sn untn = 1 and lim un vn = 1. Then multiplying these two limits together, lim sn vntn so sn vn tn. □

Sources

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

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Problems to Work

Problems to Work for Understanding

  1. Let 0 < p < 1. Let kn = np be the least integer greater than or equal to np. Then show that kn 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

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Books

Reading Suggestion:

References

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Links

Outside Readings and Links:

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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