6 Convergence Properties
6.1 Convergence Definitions
Definition
25
Convergence in Probability to Zero
A sequence of random variables \(\{ X_n\} \) converges in probability to zero if:
\[ \forall \varepsilon {\gt} 0, \quad \mathbb {P}(|X_n| {\gt} \varepsilon ) \to 0 \text{ as } n \to \infty \]
Definition
26
Convergence in Probability to a Constant
A sequence of random variables \(\{ X_n\} \) converges in probability to \(c\) if:
\[ \forall \varepsilon {\gt} 0, \quad \mathbb {P}(|X_n - c| {\gt} \varepsilon ) \to 0 \text{ as } n \to \infty \]
6.2 Known Results (Axiomatized)
The following properties capture known results from the literature that we assume as axioms:
Definition
27
Exact Gumbel Convergence Property
For a Gumbel grid and appropriate constants \(C_g, \sigma _g {\gt} 0\):
\[ \frac{T_{\text{Gumbel}}(n) - C_g \cdot n}{\sigma _g \cdot n^{1/3}} \xrightarrow {d} F_{\text{GUE}} \]
where \(F_{\text{GUE}}\) is the GUE Tracy-Widom distribution.
Definition
28
Time Constant for Gumbel LPP
There exists a constant \(D_\ell {\gt} 0\) such that:
\[ \frac{T_{\text{Gumbel}}(n)}{n} \xrightarrow {\mathbb {P}} D_\ell \]
Definition
29
Time Constant for Exponential LPP
There exists a constant \(D_L {\gt} 0\) such that:
\[ \frac{L_{\text{Exp}}(n)}{n} \xrightarrow {\mathbb {P}} D_L \]
6.3 Slutsky’s Theorem
Theorem
30
Slutsky Upper Bound
For random variables \(X, Y\) and constants \(r, \varepsilon \):
\[ \mathbb {P}(X + Y \leq r) \leq \mathbb {P}(X \leq r + \varepsilon ) + \mathbb {P}(|Y| {\gt} \varepsilon ) \]
Theorem
31
Slutsky Lower Bound
For random variables \(X, Y\) and constants \(r, \varepsilon \):
\[ \mathbb {P}(X \leq r - \varepsilon ) \leq \mathbb {P}(X + Y \leq r) + \mathbb {P}(|Y| {\gt} \varepsilon ) \]
Theorem
32
Slutsky’s Theorem for CDFs
Suppose \(X_n \xrightarrow {d} F\) (convergence in distribution to a continuous CDF \(F\)) and \(Y_n \xrightarrow {\mathbb {P}} 0\). Then:
\[ X_n + Y_n \xrightarrow {d} F \]
Proof
▶
Fix \(r \in \mathbb {R}\) and \(\varepsilon {\gt} 0\). By Theorems 30 and 31:
\begin{align*} \mathbb {P}(X_n \leq r - \varepsilon ) - \mathbb {P}(|Y_n| {\gt} \varepsilon ) & \leq \mathbb {P}(X_n + Y_n \leq r) \\ & \leq \mathbb {P}(X_n \leq r + \varepsilon ) + \mathbb {P}(|Y_n| {\gt} \varepsilon ) \end{align*}
Taking limits and using continuity of \(F\) gives the result.
6.4 Auxiliary Convergence Lemmas
Lemma
33
Product Convergence
If \(Y_n \xrightarrow {\mathbb {P}} c\) and \(a_n \to 0\), then \(a_n \cdot Y_n \xrightarrow {\mathbb {P}} 0\).
Lemma
34
Deterministic Factor Limit
For \(\alpha {\gt} 2/3\):
\[ \frac{n}{\lfloor n^\alpha \rfloor \cdot n^{1/3}} \to 0 \text{ as } n \to \infty \]
Proof
▶
We have:
\[ \frac{n}{\lfloor n^\alpha \rfloor \cdot n^{1/3}} \leq \frac{2n}{n^\alpha \cdot n^{1/3}} = 2n^{2/3 - \alpha } \]
Since \(\alpha {\gt} 2/3\), the exponent is negative and the limit is zero.