Conditional Convergence in Probability

Convergence in probability is the simplest form of convergence for random variables: for any positive ε it must hold that P[ | X_n - X | > ε ] → 0 as n → ∞. This kind of convergence is easy to check, though harder to relate to first-year-analysis convergence than the associated notion of convergence almost surely: P[ X_n → X as n → ∞] = 1.

Convergence in probability is implied by convergence almost surely (most direct proof: express convergence almost surely in terms of more elementary events using countable unions and intersections and do some simple reasoning on the result), but is not implied by it. On the other hand a sequence of random variables that converges in probability always has a sub-sequence that converges almost surely. Moreover if every sub-sequence of a sequence of random variables contains a sub-sub-sequence that converges almost surely, then the original sequence converges in probability.

While studying the application of Dirichlet forms to Markov chain Monte Carlo (developing the work of Zanella et al, 2017), the following convergence-in-probability question arose:

Question

Work on a given probability space (Ω, F, P). Suppose that random variables X₁, X₂, ... converge to X in probability.
Given a sub-σ-algebra G < F, is it the case that random variables X₁, X₂, ... converge to X in G-conditional probability?
In other words, is it the case that, for every ε>0,

If the conditioning is simply a conditioning on a single event of positive probability, then the answer is yes; consider that

so P[ A_n | B ] will converge to zero if P[ A_n ] does.

However the general answer is, of course, no. In a nutshell, consider a sequence of random variables that converges in probability but not almost surely. Condition on the entire sequence(!), thus rendering it entirely deterministic. There is a positive chance that the conditioned sequence fails to converge; and if so then it cannot converge in (conditional) probability. We now give an explicit example of a sequence that converges in probability but not almost surely, and spell out the details of why conditional convergence in probability then fails.

Example

Consider a Uniform random variable U defined on [0, 1), using the usual Lebesgue σ-algebra F. Define X₁, X₂, ... as follows: consider the ensemble of events [(k-1)2^-m, k2^-m ) for k = 1, ..., m and m= 1, 2, ... Order these and let X_n be the indicator random variable corresponding to the nth event, while X= 0.. Then P[ X_n =1]=2^-m if X_n is the indicator random variable corresponding to [(k-1)2^-m, k2^-m ), hence P[ | X_n - X | > ε ] → 0 if 0 < ε < 1. So certainly X₁, X₂, ... converge to X in probability. On the other hand, if G = F then almost surely the sequence X₁, X₂, ... contains infinitely many 1's as well as infinitely many 0's, so similarly for P[ | X_n - X | > ε | G ] = [X_n = 0], so almost surely P[ | X_n - X | > ε | G ] can never converge.

Discussion

More generally, this sort of problem arises whenever the σ-algebra G contains a random variable whose distribution is not atom-free.

Exactly the same argument shows that L_p convergence does not imply "conditional L_p convergence".

However the facts that

1. convergence in probability implies existence of almost surely convergent subsequences,
2. while convergence in probability itself is implied by existence of almost surely convergent sub-subsequences for every subsequence,

can be used to evade the issues raised here. For example, in the case of the application of Dirichlet forms to Markov chain Monte Carlo, even though convergence in probability is not preserved under conditioning, these considerations can be used to prove a strategic conditional CLT ...

Reference

Zanella, G., Bédard, M., & Kendall, W. S. (2017). A Dirichlet Form approach to MCMC Optimal Scaling. Stochastic Processes and Their Applications, to appear, 22pp. http://doi.org/10.1016/j.spa.2017.03.021

Wilfrid Kendall : 30 May 2017 10:01 |  Tags: Conditional Probability Convergence Almost Surely Convergence In Mean Convergence In Probability |  Comments (2) |  Report a problem