 Bounded second moments do not upgrade convergence in probability to convergence in L2

ID: bounded-second-moments-do-not-upgrade-convergence-in-probability-to-convergence-in-l2

([0, 1], B, Leb)

, the variables

X_{n} = n 1_{(0, 1/ n)}

converge in probability and in

L^{1}

to zero, while

∥ X_{n} ∥_{2} = 1

for every

n

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