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Let $X_n$ be a sequence of random variables in the probability spaces $(\Omega_n, \mathcal{A}_n, P_n )$ such that $X_n$ converges to $N(0,\sigma^2)$ in distribution. Does it imply that the sequence $\exp(X_n)$ converges to $1$ in probability? If yes, how can I show it?

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If $\exp(X_n)\to 1$ in probability then since $\log $ is a continuous function $X_n\to 0$ in probability. Contradicition.

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  • $\begingroup$ Thanks a lot ;) $\endgroup$ – inbrevi Mar 9 at 19:14

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