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
Asymptotic Limits
Rating
Mathematically Mature: may contain mathematics beyond calculus with
proofs.
1
Section Starter Question
Name some sequences sn and tn such that lim sn = ∞ and lim tn = ∞ and
lim
n→∞
sn
= 1.
tn
What does this say about the rate at which sn and tn approach ∞ ?
Key Concepts
1. For sequences sn and tn , we say that sn is asymptotic to tn , written
sn ∼ tn , if limn→∞ stnn = 1.
Vocabulary
1. For sequences sn and tn , we say that sn is asymptotic to tn , written
sn ∼ tn , if limn→∞ stnn = 1.
2
Mathematical Ideas
Definition. For sequences sn and tn , we say that sn is asymptotic to tn ,
written sn ∼ tn , if limn→∞ stnn = 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
sn un ∼ tn vn .
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 limn→∞ sn = 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(9/10)/ ln(tn ), − ln(11/10)/ ln(tn )) < η
for n > N1 . Choose N2 so large that
tn · (9/10) < sn < tn · (11/10).
Hence
ln(tn ) + ln(9/10) < ln(sn ) < ln(tn ) + ln(11/10).
Then
1+
ln(sn )
ln(11/10)
ln(9/10)
>
>1+
.
ln(tn )
ln(tn )
ln(tn )
For n > max(N1 , N2 )
1−η <
ln(sn )
<1+η
ln(tn )
so ln(sn ) ∼ ln(tn ).
Lemma 6 (Multiplication II). If sn · un ∼ tn and un 6= 0, 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 unsn·tn = 1 and lim uvnn = 1. Then
multiplying these two limits together, lim vns·tn n so sn ∼ vn · tn .
3
Sources
This section is just simple analysis of limits organized into elementary lemmas
for later reference.
Problems to Work for Understanding
1. Let 0 < p < 1. Let kn = dnpe 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
Reading Suggestion:
References
Outside Readings and Links:
1.
4
2.
3.
4.
I check all the information on each page for correctness and typographical
errors. Nevertheless, some errors may occur and I would be grateful if you would
alert me to such errors. I make every reasonable effort to present current and
accurate information for public use, however I do not guarantee the accuracy or
timeliness of information on this website. Your use of the information from this
website is strictly voluntary and at your risk.
I have checked the links to external sites for usefulness. Links to external
websites are provided as a convenience. I do not endorse, control, monitor, or
guarantee the information contained in any external website. I don’t guarantee
that the links are active at all times. Use the links here with the same caution as
you would all information on the Internet. This website reflects the thoughts, interests and opinions of its author. They do not explicitly represent official positions
or policies of my employer.
Information on this website is subject to change without notice.
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
5