2022-Q4 — Upper Bounds for Brownian Motion Maxima
2022 Probability Prelim Exam (PDF)
Theorems & Tools Used (Cheat-Sheet Index)
- Markov’s Inequality:
$\mathbb P(X>a)\le \mathbb E[X]/a$ for $X\ge 0$. - Chernoff (Exponential Markov) Bound:
Apply Markov to $e^{tX}$, then optimize over $t>0$. - Normal MGF:
If $Z\sim N(0,1)$, then $\mathbb E[e^{tZ}]=e^{t^2/2}$. - Reflection Principle (Brownian Motion):
$\mathbb P(\max_{s\le t}B(s)>x)=2\mathbb P(B(t)>x)$. - Brownian Scaling:
$B(t)\stackrel{d}{=}\sqrt t\,Z$, $Z\sim N(0,1)$. - Borel–Cantelli Lemma (I):
If $\sum_n \mathbb P(A_n)<\infty$, then $\mathbb P(A_n\ \mathrm{i.o.})=0$. - $p$-Series Test:
$\sum_{n=1}^\infty n^{-p}<\infty$ iff $p>1$. - Monotonicity of $\log\log x$ (eventually):
Used to compare normalizing terms on intervals. - Limit Rules for Limsup:
$\limsup (a_nb_n)\le (\limsup a_n)(\limsup b_n)$ for $a_n,b_n\ge 0$.
Verbatim Question (from Prob_F22)
Let $Z \sim N(0,1)$.
(a)(i) Prove that \(\mathbb{P}(Z>x) \le e^{-x^2/2}, \qquad x>0.\)
(a)(ii) Let $(B(s))_{s\ge0}$ be standard Brownian motion. Prove that \(\mathbb{P}\!\left(\max_{0\le s\le t} B(s) > x\right) \le 2\exp\!\left(-\frac{x^2}{2t}\right), \qquad x>0,\ t>0.\)
For the rest of the problem, let $\alpha>1$ and define $t_n=\alpha^n$, $n=1,2,\dots$.
(b) Show that \(\limsup_{n\to\infty} \frac{\max_{0\le s\le t_n} B(s)} {\sqrt{2t_{n+1}\log\log(t_{n+1})}} \le 1 \quad\text{a.s.}\)
(c) Show that \(\limsup_{n\to\infty} \max_{t_{n-1}\le s\le t_n} \frac{B(s)}{\sqrt{2s\log\log(s)}} \le \alpha \quad\text{a.s.}\)
(a)(i)
Claim
For $Z\sim N(0,1)$ and $x>0$, \(\mathbb{P}(Z>x)\le e^{-x^2/2}.\)
Proof
For any $t>0$, \(\mathbb{P}(Z>x)=\mathbb{P}(e^{tZ}>e^{tx}) \le \frac{\mathbb{E}[e^{tZ}]}{e^{tx}} = \exp\!\left(\frac{t^2}{2}-tx\right),\) by Markov’s inequality applied to $e^{tZ}$ (Chernoff bound) and the normal MGF. Minimizing the exponent over $t>0$ gives $t=x$, yielding the result.
Conclusion
The standard normal upper tail is sub-Gaussian with variance proxy $1$.
Key Takeaway
Chernoff bounds plus MGF optimization give sharp Gaussian tail estimates.
(a)(ii)
Claim
For $x>0$ and $t>0$, \(\mathbb{P}\!\left(\max_{0\le s\le t} B(s)>x\right) \le 2\exp\!\left(-\frac{x^2}{2t}\right).\)
Proof
By the reflection principle, \(\mathbb{P}\!\left(\max_{0\le s\le t} B(s)>x\right) =2\,\mathbb{P}(B(t)>x).\) Using Brownian scaling, $B(t)=\sqrt t\,Z$ with $Z\sim N(0,1)$, and applying part (a)(i) gives the bound.
Conclusion
Finite-time Brownian maxima inherit Gaussian tail decay.
Key Takeaway
Reflection principle + scaling reduces pathwise bounds to one-dimensional Gaussian tails.
(b)
Claim
If $\alpha>1$ and $t_n=\alpha^n$, \(\limsup_{n\to\infty} \frac{\max_{0\le s\le t_n} B(s)} {\sqrt{2t_{n+1}\log\log(t_{n+1})}} \le 1 \quad\text{a.s.}\)
Proof
Define \(A_n=\left\{M_{t_n}>\sqrt{2t_{n+1}\log\log(t_{n+1})}\right\}.\) Using (a)(ii), \(\mathbb{P}(A_n)\le 2\exp\!\left(-\alpha\log\log(t_{n+1})\right) \le C(n+1)^{-\alpha}.\) Since $\alpha>1$, the series $\sum_n \mathbb{P}(A_n)$ converges by the $p$-series test. By Borel–Cantelli (I), $A_n$ occurs only finitely often, giving the almost-sure bound.
Conclusion
On a geometric time grid, Brownian maxima are eventually controlled by the LIL scale.
Key Takeaway
Borel–Cantelli turns summable tail bounds into almost-sure eventual control.
(c)
Claim
\(\limsup_{n\to\infty} \max_{t_{n-1}\le s\le t_n} \frac{B(s)}{\sqrt{2s\log\log(s)}} \le \alpha \quad\text{a.s.}\)
Proof
By monotonicity of $\sqrt{2s\log\log s}$, \(\max_{t_{n-1}\le s\le t_n} \frac{B(s)}{\sqrt{2s\log\log(s)}} \le \frac{M_{t_n}}{\sqrt{2t_{n-1}\log\log(t_{n-1})}}.\) Use the eventual bound from part (b) and compute the limit of the ratio of normalizing constants to obtain $\alpha$.
Conclusion
Blockwise Brownian fluctuations obey an LIL upper bound with constant $\alpha$.
Key Takeaway
Discrete-time almost-sure bounds extend to continuous intervals via deterministic scale comparisons.
Summary of Key Takeaways
- Markov + Chernoff + MGFs control Gaussian tails.
- Reflection principle is essential for Brownian maxima.
- Borel–Cantelli converts summable probabilities into a.s. bounds.
- Geometric discretization ($t_n=\alpha^n$) avoids critical divergence.
- $p$-series convergence ($p>1$) is a recurring tool in LIL proofs.
- The $(1+\varepsilon)$-trick is only needed at critical scaling.
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