# Convergence in L2 (up to a constant) implies convergence in probability?

Suppose we have a sequence of random variables $${X_n}$$ such that $$mathbb{E} [|X_n|^2] = a_n + c$$ where $$a_n to 0$$ is a decreasing sequence of positive real numbers and $$c > 0$$ is a constant. Then can we say anything about the convergence in probability of $$X_n$$?

Mathematics Asked by user3294195 on November 18, 2021

No. Consider the sequence of random variables $$X_n = (1 + frac{1}{n})$$ with the uniform measure on $$[0,1]$$. Well $$mathbb{E}(|X_n|^2) = frac{1}{n^2} + frac{2}{n} + 1$$ where $$c = 1$$ and $$a_n = frac{1}{n^2} + frac{2}{n}$$ fulfill your properties. Now consider the sequence of random variables $$Z_n = (-1)^n X_n$$. You get that $$Z_n$$ have the same properties, but $$Z_n$$ does not converge in measure/probability to any random variable; the two candidates being $$1$$ and $$-1$$.

Answered by Andrew Shedlock on November 18, 2021

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