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$L^p$ Convergence

\(X_n \xrightarrow[n\to\infty]{L^p} X \qquad (p \ge 1)\)

Definition

A sequence $X_n$ converges to $X$ in $L^p$ if \(\mathbb{E}\!\left[ |X_n - X|^p \right] \to 0.\)

This defines a metric convergence in the Banach space $L^p$.

Interpretation

$L^p$ convergence means:

The $p$-th moment of the error $X_n - X$ vanishes.

It is a strong, quantitative notion of convergence that controls both magnitude and probability of deviations.

Heuristically:

  • $L^1$: average error goes to zero
  • $L^2$: mean-square error goes to zero

Relationship to Other Modes of Convergence

For $p \ge 1$, \(X_n \xrightarrow{L^p} X \;\Rightarrow\; X_n \xrightarrow{p} X.\)

For $p > q \ge 1$, \(X_n \xrightarrow{L^p} X \;\Rightarrow\; X_n \xrightarrow{L^q} X.\)

$L^p$ convergence does not imply almost sure convergence in general.

How $L^p$ Convergence Typically Arises

1. Dominated Convergence

If

  • $X_n \to X$ a.s., and
  • $ X_n - X ^p \le Y$ with $\mathbb{E}[Y] < \infty$,

then \(X_n \xrightarrow{L^p} X.\)

This is the most common mechanism for proving $L^p$ convergence.

2. Uniform Integrability (for $p=1$)

If

  • $X_n \to X$ in probability, and
  • ${ X_n }$ is uniformly integrable,

then \(X_n \xrightarrow{L^1} X.\)

This is a standard route in martingale and empirical process theory.

3. Variance Control (for $p=2$)

If \(\mathbb{E}\!\left[(X_n - X)^2\right] \to 0,\) then \(X_n \xrightarrow{L^2} X.\)

This is common in Hilbert space arguments and orthogonal decompositions.

Special Case: $L^1$ Convergence

\[X_n \xrightarrow{L^1} X \quad \Longleftrightarrow \quad \mathbb{E}|X_n - X| \to 0.\]

Interpretation

  • Controls expected absolute error.
  • Guarantees convergence of expectations: \(\mathbb{E}[X_n] \to \mathbb{E}[X].\)

Key Tools

  • Uniform integrability,
  • Dominated convergence,
  • Martingale convergence in $L^1$.

Special Case: $L^2$ Convergence

\[X_n \xrightarrow{L^2} X \quad \Longleftrightarrow \quad \mathbb{E}[(X_n - X)^2] \to 0.\]

Interpretation

  • Controls mean squared error.
  • Natural geometry: inner product space.

Useful Identity

\[\mathbb{E}[(X_n - X)^2] = \mathrm{Var}(X_n - X) + (\mathbb{E}[X_n - X])^2.\]

This decomposition is often used in proofs.

Key Tools

  • Orthogonality and projections,
  • Pythagorean identity,
  • Conditional expectation as an $L^2$ projection.

Common Pitfalls

  • $X_n \xrightarrow{p} X$ does not imply $L^p$ convergence.
  • Almost sure convergence alone is insufficient for $L^p$ convergence.
  • $L^p$ convergence requires moment control.

One-line Summary

$L^p$ convergence means the $p$-th moment of the error goes to zero; it implies convergence in probability and provides quantitative control, with $L^1$ governing expectations and $L^2$ governing mean-square error.

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