Final answer:
The provided statement is false; convergence of the conditional expectation to 0 in probability does not ensure that the sequence of random variables itself converges to 0 in probability.
Step-by-step explanation:
The statement 'If the conditional expectation of a sequence of random variables converges to 0 in probability, then the random variables converge to 0 in probability' is false. Convergence of the conditional expectation to 0 in probability does not necessarily imply the convergence of the sequence itself to 0 in probability. The conditional expectation might converge due to the law of iterated expectations, where it reflects some average property of the sequence rather than the behavior of individual terms. To claim convergence in probability of the random variables themselves, one must show that for any ε > 0, the probability that the absolute difference between the random variable and 0 is less than ε, approaches 1 as the sequence progresses. This direct comparison to 0, rather than to conditional expectations, is crucial in establishing the convergence in probability of the sequence of random variables themselves.
The information provided regarding the central limit theorem and the law of large numbers is related but does not directly address the convergence of a sequence of random variables to a point, hence it is not pertinent to the question at hand.