# The continuity assumption on the limiting c.f. in Levy's continuity theorem.

I am looking for an example that shows we can't drop the continuity assumption on the limiting characteristic function in Levy's continuity theorem.

I got some hints from my professor that use a sequence of normal distributions {$\mu_n$ } with $\mu_n$ having mean 0 and variance n.

Could anyone help me out with how this construction works?

• So what is the limit of the characteristic functions of the $\mu_n$ in the professor's hint. Is it continuous? Is it the sequence $\mu_n$ tight? – kimchi lover Aug 3 '17 at 22:51
• Why can't you follow the precise suggestion by your teacher? (Unrelated: Are you still using two accounts on mse?) – Did Aug 6 '17 at 21:36