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manpreet Best Answer 2 years ago

There is a n">question in a syllabus called "Mathematische statistiek":

"If every $X_{n}$ and $X$ possess a discrete distribution supported on a finite set of integers, show that $X_{n}$ converges in distribution to $X$ iff $P (X_{n} = x) \to P (X = x)$ for every $x$ ."

This is what I tried: First assume the convergence holds. Let $k_{1} < k_{2} < . . . < k_{n}$ be all the integers $X$ can reach. For $i \in N - {1}$ we have

P (X n \leq k i) - P (X REPLY 0 views 0 likes 0 shares Facebook Twitter Linked In WhatsApp

manpreet 2 years ago

P (X n = k i) = P (X n \leq k i + 0.1) - P (X n \leq k i - 0.1) \to \to P (X \leq k i + 0.1) - P (X \leq k i - 0.1) = P (X = k i)

since the unique integer value

k_{i}

belongs to the interval

(k_{i} - 0.1, k_{i} + 0.1]

. The convergence takes place since CDF of

X

is continuous at the points

k_{i} \pm 0.1

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Weak convergence with equilities instead of inequilities

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manpreet Best Answer 2 years ago

manpreet 2 years ago

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