The unit price of a certain commodity evolves randomly from day to day with a general downward drift but with an occasional upward jump when some unforeseen event excites the markets. Long term records suggest that, independently of the past, the daily price increases by a dollar with probability $0.45$, declines by $2$ dollars with probability $0.5$, but jumps up by $10$ dollars with probability $0.05$. Let $C_0$ be the price today and $C_n$ the price $n$ days into the future. How does the probability $P(C_n>C_0)$ behave as $n$ goes to infinity?
My work:
Consider $\lim_{n \to \infty} (C_n>C_0)$ where $Cn=\frac{\Sigma_{i = 1}^{n}X_i}{n}$
where $X_i$ are i.i.d. random variables. Let $\mu$ be the mean of $X_i$ and $\sigma$ be the standard deviation.
$\lim_{n \to \infty} P(C_n>C_0)=1-P(C_n＜C_0) ＜1-P(|C_{n-m}|＜C_0) ＜(1-(1-\frac{sigma^2}{nC_0^2}))=0$
Suppose $D_n = C_n - C_0$. Then $P(C_n > C_0) = P(D_n > 0) = P(\frac{D_n}{n} > 0)$, and $D_n = \Sigma_{k = 1}^{n} X_i$, where $X_i$ are i.i.d. random variables, such that $P(X_i = 1) = 0.45$, $P(X_i = -2) = 0.5$ and $P(X_i = 10) = 0.05$. Then by the Law of Large Numbers $\frac{D_n}{n}$ converges in probability to $EX_i = 0.45*1 + 0.5*(-2) + 0.05*10 = -0.05 < 0$. That results in $P(C_n > C_0) = P(\frac{D_n}{n} > 0) = 0$
You can find more about convergence in probability here: https://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_probability