Convergence in probability

Depends on

• Convergence in measure
• Probability measures and probability spaces
• Random elements and real random variables

Used by

• Deterministic Ito integrals are Gaussian Corollary
• The Brownian square martingale Corollary
• A macroscopic row term defeats averaging Counterexample
• An unbounded stopped exponential martingale needs uniform integrability Counterexample
• Convergence in distribution need not be convergence in probability Counterexample
• Convergence in probability need not be almost sure Counterexample
• Convergence in probability need not imply Lᵖ convergence Counterexample
• The uniform endpoint MLE has rate n, not square-root n Counterexample
• Weak law does not imply strong law Counterexample
• Quadratic covariation of Brownian Ito processes Definition
• Stochastic order and convergence notation for finite-dimensional random vectors Definition
• A probability-convergent sequence with a prescribed fast almost-sure subsequence Example
• Hitting probabilities from an exponential martingale Example
• Normal location fitted under the wrong variance Example
• Cauchy sequences in probability have a measurable limit Lemma
• Characteristic exponential for a continuous local martingale with deterministic clock Lemma
• Finite-alphabet IID information AEP with a variance bound Lemma
• Finite-data fixed-time NTK stability under integrably bounded training directions Lemma
• Local-linear moment matrix converges to a positive-definite limit Lemma
• Pairing preserves convergence in probability Lemma
• Scaled Plancherel profile moments converge in probability Proposition
• Central-limit convergence is only in distribution Remark
• Almost-sure convergence implies convergence in probability Theorem
• An almost-surely convergent subsequence from convergence in probability Theorem
• Brownian-filtration martingale representation Theorem
• Chebyshev weak law for uncorrelated arrays Theorem
• Conditional asymptotic variance of Nadaraya–Watson and local-linear estimators Theorem
• Conditional-on-design bias of the empirical local-linear estimator Theorem
• Continuous maps preserve convergence in probability Theorem
• Convergence in distribution to a constant is convergence in probability Theorem
• Convergence in probability and almost surely agree for independent series Theorem
• Convergence in probability implies convergence in distribution Theorem
• Convergence in probability is metrized by d₀ Theorem
• Converging together lemma Theorem
• Exact tail criterion for a truncated-centered IID weak law Theorem
• Fixed-order autoregressive OLS martingale limit Theorem
• Likelihood-ratio limit for local cones in a QMD model Theorem
• Limits in probability are unique almost surely Theorem
• Lᵖ convergence implies convergence in probability Theorem
• Misspecified M-estimator sandwich limit Theorem

…and 9 more results.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Rick Durrett, Probability: Theory and Examples, 5th ed., Section 2.2 (standard reference, not scraped)