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cwbullivant
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Homework Statement
Let $$ \it{f}(x) $$ be a probability density function. Now let Xn have the density:
$$ \it{f}_{n}(x) = n\it{f}(nx) $$
Determine whether or not Xn converges in distribution to zero.
(this is the verbatim statement, there is no additional information given)
Homework Equations
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If the CDF $$ F_{n}(X) \rightarrow 0, n \rightarrow \infty $$ then Xn converges to zero in distribution.
The Attempt at a Solution
Definition of CDF:
$$ F_{n}(x) = P (X < x) = \int_{-\infty}^{x} \it{f}_{n}(x) dx= \int_{-\infty}^{x} n\it{f}(nx) dx $$
And to be a valid PDF, its integral from -∞ to +∞ must be 1. That requires that f(nx) must go to zero as nx goes to ±∞. So the lower limit on that integral is already taken care of; that part has to vanish if nf(nx) is going to be a valid PDF.
But for finite x, wouldn't the behavior of f(nx) as n goes to ∞ depend on exactly what f(nx) looks like (e.g. would go to zero if f(nx) = e^{-nx}, but not if f(nx) = ne^(-x))? I'm not sure on where to go from here.