Asymptotic Notation
Big-O, little-o, and tilde (~) notation for describing behavior at infinity.
When studying fat tails, we care about how functions behave as becomes very large. Asymptotic notation gives us precise language for comparing functions at infinity — crucial for understanding tail behavior.
Asymptotic Equivalence (~)
Asymptotic Equivalence
We write as if:
Read: " is asymptotically equivalent to "
Polynomial Asymptotics
Consider . As :
So . The lower-order terms become negligible.
Stirling's Approximation
One of the most famous asymptotic formulas:
For large , the right side approximates with increasing accuracy. At , the relative error is less than 1%.
Why Asymptotics Matter for Fat Tails
We often write tail behavior as:
This says: for large , the survival probability behaves like a power law with exponent . The constant and lower-order corrections don't change the fundamental tail behavior.
Big-O Notation: Upper Bounds
Big-O
We write as if there exist constants and such that:
Read: " is big-O of " or " is at most order "
Big-O gives an upper bound on growth. It says " grows no faster than (up to a constant)".
Common Big-O Statements
- — the term dominates
- — sine is bounded by a constant
- for any — logs grow slower than any power
- — factorial grows slower than
Caution: Equals Sign is Misleading
The notation is not symmetric! We can't write. Think of it as " is a member of the set of functions bounded by ".
Little-o Notation: Strictly Smaller
Little-o
We write as if:
Read: " is little-o of " or " is negligible compared to "
Little-o is stronger than big-O. While means is at most comparable to , means becomes negligible compared to .
Comparing O and o
- ✓ and ✓ — grows slower than
- ✓ but — same order, not negligible
- for any — exponential decay beats any polynomial decay
Read: “f equals little-o of g”
f becomes negligible compared to g; the ratio f/g approaches zero
Explore: Comparing Growth Rates
Visualize how different functions grow and see the ratio f(x)/g(x) approach 0.
f(x) and g(x)
Both functions grow, but at different rates
Ratio f(x) / g(x)
Ratio approaches 0 — this is what o(·) means
Statement:
ln(x) = o(x)
f(100) / g(100):
0.046052
Interpretation:
f is small compared to g
ln(x) vs x
Logarithmic growth is much slower than linear. No matter how large x gets, ln(x)/x → 0.
Asymptotic Notation Summary
f = o(g)
f/g → 0
f is negligible
f = O(g)
|f/g| ≤ C
f bounded by g
f ~ g
f/g → 1
f ≈ g for large x
Comparing the Notations
| Notation | Definition | Meaning | Example |
|---|---|---|---|
| Same asymptotic behavior | |||
| f grows at most as fast as g | |||
| f grows strictly slower than g |
Asymptotic Notation in Fat Tail Theory
Throughout this course, you'll see asymptotic notation used to describe tail behavior:
Pareto Tails
The survival function is asymptotically a power law with exponent .
Subexponential Definition
For subexponential distributions, the probability that a sum is large is asymptotically twice the probability that a single variable is large.
CLT Rate of Convergence
The Berry-Esseen theorem: convergence to normal is at rate .
Reading Asymptotic Statements
When you see in this course:
- The statement is about large x (or large n)
- and become essentially equal (in ratio)
- Lower-order terms are absorbed into the approximation
- We care about the dominant behavior, not exact values
Preview: Slowly Varying Functions
A concept we'll explore in depth later is the slowly varying function, which plays a key role in fat tail theory.
Slowly Varying Function
A function is slowly varying at infinity if:
Examples include constants, , , and. These grow, but so slowly that scaling by any constant doesn't change the ratio asymptotically.
Why "Slowly"?
Compare to a power :
- , ,
- For : , ,
The log grows, but compared to any power (even tiny ones), it's negligible: for any .
When we write , the slowly varying captures deviations from a pure power law without changing the fundamental tail behavior.
Explore: Slowly Varying Functions
Verify that L(tx)/L(x) → 1 for slowly varying functions, regardless of the scaling factor t.
L(x) and L(2x)
Both functions grow, but the gap between them stabilizes
Ratio L(2x) / L(x)
Ratio approaches 1 as x increases
Function:
L(x) = ln(x)
Ratio at x = 1000:
L(2×1000) / L(1000) = 1.1003
Compare to Power Functions
For a power function g(x) = xα, the ratio is:
g(tx) / g(x) = (tx)α / xα = tα
This stays constant at tα — it doesn't approach 1. That's the key difference: slowly varying functions "don't care" about multiplicative scaling.
Key Takeaways
- : and are asymptotically equivalent ()
- : is bounded by a constant times
- : becomes negligible compared to ()
- Fat tail statements like describe behavior for large
- Slowly varying functions grow slower than any power and appear in refined tail descriptions