Asymptotic (limsup) Growth ⇒ Global Bounds
Core Pattern
Any assumption of the form
\(\limsup_{|x|\to\infty} \frac{|g(x)|}{h(x)} < \infty\)
provides tail control only.
To use it in integrals or expectations, it must be converted into a global bound.
Standard Conversion (Template)
Assume:
- $ \limsup_{|x|\to\infty} |g(x)|/h(x) \le C $,
- $ g $ is continuous (or locally bounded),
- $ h(x)\ge 0 $.
Then:
-
Tail control
There exists $R>0$ and $C_1>0$ such that \(\|x\|>R \;\Rightarrow\; |g(x)| \le C_1 h(x).\) -
Compact control
By continuity, $g$ is bounded on $|x|\le R$: \(|g(x)| \le M \quad \text{for } |x|\le R. mn,klj,n\) -
Global envelope (enlarge constants)
There exists $C’=\max{M,C_1}$ such that \(|g(x)| \le M+C_1h(x) = C'(1+h(x)) \quad \forall x.\)
This step absorbs all constants and removes piecewise bounds.
Canonical Examples
Polynomial growth \(\limsup_{|x|\to\infty} \frac{|f(x)|}{|x|^k} < \infty \quad\Rightarrow\quad |f(x)| \le C(1+|x|^k).\)
Exponential growth \(\limsup_{|x|\to\infty} \frac{|f(x)|}{e^{a|x|}} < \infty \quad\Rightarrow\quad |f(x)| \le C(1+e^{a|x|}).\)
Tail probabilities \(\limsup_{x\to\infty} x^\alpha P(|X|>x) < \infty \quad\Rightarrow\quad P(|X|>x) \le \frac{C}{x^\alpha} \text{ for large } x.\)
Why This Works
- limsup assumptions control only infinity,
- continuity controls compact sets,
- $1+h(x)$ merges both regimes,
- constants are allowed to change.
Exam-Safe Justification (One Line)
“The limsup condition controls the tails, continuity gives boundedness on compact sets, and enlarging constants yields a global bound.”
What This Buys You (Gaussian Context)
If $Z\sim N(0,1)$ and $h(x)=|x|^k$:
- $E|g(Z)|<\infty$,
- boundary terms $g(x)\phi(x)\to 0$,
- integration by parts is justified.
Mental Trigger (Do Not Memorize Formulas)
Asymptotic control ⇒ split into compact + tails ⇒ absorb constants ⇒ global bound.
This pattern applies broadly in:
- Gaussian IBP / Stein identities
- Moment and tail arguments
- Borel–Cantelli setups
- Uniform integrability checks
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