1
vote
Accepted
Finiteness of hitting 1 for asymmetric random walk using martingales - uniform integrability
When $0<p<1/2$ and $t>0$, $0<\phi_X(t)<1$ for small positive $t$. For such $t$, your argument runs into a problem because $\phi_X(t)^{-T}=+\infty$ on the event $\{T=\infty\}$.
But ...
- 28.4k
1
vote
Showing $\limsup \frac{\mid X_n \mid}{\sqrt{n}} < K$ a.s if it is finite a.s. and $(X_n)_n$ are iidrv
Let $Y=\limsup \frac{\mid X_n \mid}{\sqrt{n}}$. Then $0 \le Y <\infty$ with probability $1$. But $P(Y<K) \to 1$ as $ K \to \infty$ [which is true for any real valued random variable $Y$].
If $P(...
- 41.6k
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