I have question about example illustration Convergence of random variables in probability but not almost surely.
Suppose a person takes a bow and starts shooting arrows at a target. Let $X_n$ be his score in $n$-th shot. Initially he will be very likely to score zeros, but as the time goes and his archery skill increases, he will become more and more likely to hit the bullseye and score $10$ points. After years of practice the probability that he hit anything but $10$ will be getting increasingly smaller and smaller and will converge to $0$. Thus, the sequence $X_n$ converges in probability to $X = 10$.
Note that $X_n$ does not converge almost surely however. No matter how professional the archer becomes, there will always be a small probability of making an error. Thus the sequence $(X_n)$ will never turn stationary: there will always be non-perfect scores in it, even if they are becoming increasingly less frequent.
For me this example is true. But not for 69.181.249.190. Why? You have any idea?
69.181.249.190 talk 36,722 bytes -957 →Convergence in probability: removed false archer example
Best Answer
The archer example incorrectly suggests that if there is always a small-but-positive probability of missing the target, then the archer misses infinitely often.
Counter-example: Consider $\{X_n\}_{n=1}^{\infty}$ with: $$X_n=\left\{\begin{array}{} 10 & \mbox{ with prob $1-1/2^n$}\\ 0& \mbox{ else (miss target)} \end{array}\right.$$ We don’t care about independence but you can assume the variables are mutually independent if you want. We observe $$\sum_{n=1}^{\infty}P[X_n\neq 10]<\infty$$ So by the Borel-Cantelli lemma we conclude that (with prob 1) we miss only finitely often. So $X_n\rightarrow10$ both in probability and with prob 1.
On the other hand, if the variables are mutually independent with $$P[X_n=10]=1-1/n, P[X_n=0]=1/n$$ then we can conclude (either by Borel-Cantelli or by direct calculation) that we miss infinitely often with prob 1. This is a classic example where $X_n\rightarrow10$ in probability but not with prob 1.
Overall, it depends on the rate at which the miss probabilities converge to zero.