Stochastic convergence is the convergence of R.V. & Distributions.

Convergence concepts are build on topologies, which is commonly specified by a metric on a metric space.

Convergence mode | notation | $( \mathcal{L}, d(\cdot,\cdot) )$ |
---|---|---|

Sure conv. | $e$ | $(\mathbb{R},\mid\cdot\mid)$ , pointwise in sample space |

Almost sure conv. | $a.e.$ | $(\mathbb{R},\mid\cdot\mid)$ , except for a measure-zero set |

Probability | $p$ | $(\mathcal{L}_p, \mathbb{E} \tfrac{\mid\cdot\mid}{1+\mid\cdot\mid} )$ |

$l_r$ | $l_r$ | $(\mathcal{L}_r,\lVert\cdot\lVert r )$ |

$l_1$ | $l_1$ | $(\mathcal{L}_1,\lVert\cdot\lVert 1)$ |

$l_2$ | $l_2$ | $(\mathcal{L}_2,\lVert\cdot\lVert 2)$ |

A simple sufficient condition for almost sure convergence is: $$\forall \varepsilon >0, \sum_{n=1}^{\infty} P{ |X_n-X| \geq \varepsilon } < \infty$$

Proof:

$$
\begin{align*}
& X_n \xrightarrow{a.e.} X \\
\iff & P{ \lim_{n \to \infty} X_n(\omega) \neq X(\omega) } =0 \\
\iff & \forall \varepsilon >0, P{ lim_{n \to \infty} \cup_{m\geq n} { |X_m-X| \geq \varepsilon } } =0 \\
\iff & \forall \varepsilon >0, lim_{n\to \infty} P{ \cup_{m\geq n} { |X_m-X| \geq \varepsilon } } =0 \\
\Longleftarrow & \forall \varepsilon >0, lim_{n\to \infty} \sum_{m\geq n} P{ |X_m-X| \geq \varepsilon } = 0 \\
\iff & \forall \varepsilon >0, \sum_{n=1}^{\infty} P{ |X_n-X| \geq \varepsilon } < \infty
\end{align*}
$$

The typical definition of convergence in probability is: $$\forall a>0, \forall \varepsilon>0, \exists N\in \mathbb{N}: \forall n>N, P{ |X_n-X| \geq a } < \varepsilon$$

An alternative definition is based on metric space $\mathcal{L}_p$, where $\mathcal{L}_p$ is the space of r.v.'s with finite norm $\mathbb{E}\tfrac{|\cdot|}{1+|\cdot|}$, and the metric is the one associated with the norm. $\mathcal{L}_p$ space is complete.

It can be shown using basic inequalities that the two definitions are equivalent.

**Stochastic order notation**

We denote $X_n = o_p(1)$, if the sequence of random variables ${ X_n }$ **converges to 0 in probability**.
Symbolically, $X_n \overset{p}{\to} 0$.

We denote $X_n = O_p(1)$, if the sequence of random variables ${ X_n }$ is **uniformly bounded in probability**.
Symbolically, $$\forall \varepsilon >0, \exists M>0: \limsup_{n\to\infty} P \{ \lvert X_n \rvert >M \} \leq \varepsilon$$

It can be proved that manipulating rules for order in probability notation is in direct parallel to other big-O notations. For example,

$$o_p(1) + o_p(1) = o_p(1), \quad o_p(1) + O_p(1) = O_p(1), \quad O_p(1) + O_p(1) = O_p(1) \\ o_p(1) o_p(1) = o_p(1), \quad o_p(1) O_p(1) = o_p(1), \quad O_p(1) O_p(1) = O_p(1)$$

The $\mathcal{L}_r$-norm of r.v. is $||\cdot||_r = ( \mathbb{E}|\cdot|^r )^{\frac{1}{r}}$

The $\mathcal{L}_r$ space of r.v.'s is *almost* a metric space, where the metric $L_r$-norm satisfies **nonnegativity**, **symmetry**, and **triangle inequality** (Minkowski's inequality), but **positivity** is not satisfied.

If we treat equivalence classes $[X] = \{ Y \in \mathcal{L}_r \mid Y=X \text{ almost surely} \}$ as fundamental points in the $\mathcal{L}_r$ space, then the $\mathcal{L}_r$ space becomes a metric space.

It can be shown that $\mathcal{L}_r (\mathcal{U},\mathcal{F},P)$ is complete, and $\mathcal{L}_2 (\mathcal{U},\mathcal{F},P)$ is a Hilbert space.

Hierarchy of r.v. spaces: $$\mathcal{L}_p \supsetneq \mathcal{L}_1 \supsetneq \mathcal{L}_2 \supsetneq \cdots \supsetneq \mathcal{L}_{\infty}$$

Convergence of distribution functions is pointwise convergence to some distribution function, except for its discontinuous points. The underlying metric space is $([0,1],|\cdot|)$. Denoted as $\mathbf{X}_n \Rightarrow \mathbf{X}$.

The corresponding sequance of random variables is said to **converge weakly**.

Note:

- It's possible for a sequence of distribution functions to converge pointwise to a function that is not a distribution function.
- Other definitions are weak convergence and convergence is Levy metric. The three definitions can be shown to be identical.
- The (perhaps) only general theorem on asymptotic distribution is the CLTs, which is based on sample average, or at least the average of a series of random variables. Other types of sequences of r.v.'s do not have an easy result for their limiting distribution.

A series of random vectors converges in distribution iff all its one-dimensional projection also converges in distribution.

Definition: (the **portmanteau theorem**)
Let S be a metric space with its Borel σ-algebra Σ. We say that a sequence of probability measures ${ P_n }$ on (S, Σ) converges weakly to the probability measure P,
if and only if any of the following equivalent conditions is true:

- $\mathbb{E}_n f \Rightarrow \mathbb{E} f$ for all bounded, continuous functions f;
- $\mathbb{E}_n f \Rightarrow \mathbb{E} f$ for all bounded and Lipschitz functions f;
- $\limsup \mathbb{E}_n f \leq \mathbb{E} f$ for every upper semi-continuous function f bounded from above;
- $\liminf \mathbb{E}_n f \geq \mathbb{E} f$ for every lower semi-continuous function f bounded from below;
- $\limsup P_n (C) \leq P(C)$ for all closed sets C of space S;
- $\liminf P_n (U) \geq P(U)$ for all open sets U of space S;
- $\lim P_n (A) = P(A)$ for all continuity sets A of measure P.

In the common case when $S=\mathbb{R}$ with its usual topology, then weak convergence of probability measures is equivalent to convergence of distribution functions.

Weak convergence of probability measures is denoted as $P_n \rightarrow P$.

Relations of convergence modes

(Proof)

Neither of them implies the other.

Convergence of expectation is defined as: $$\lim_{n\to\infty} \mathbb{E}[Z_n - Z] =0$$

Convergence of expectation is related to convergence in $L_1$, but much weaker. Still, it's not weaker than convergence in probability. Counterexamples typically have unbounded heavy tail. Consider sequence $Z_n$ which takes value n with probability $\frac{1}{n}$, and equals 0 otherwise. This sequence converges to degenerated r.v. 0 in probability, but converges to 1 in expectation.