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Results tagged with random-walks
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user 49551
2
votes
1
answer
421
views
Random walk always stays below a level $a$
Suppose we have a random walk $S_n$ with i.i.d. steps $X_i$. We assume that
$$\mathbb{E}[X_i] = -\mu, \text{Var}[X_i] = 1,$$
where $\mu$ is close (or going) to zero. We also assume that the moment gen …
- 485
6
votes
0
answers
146
views
Running minimum of exponential random walks
Let $\{X_i\}$ be a collection of i.i.d. Exp$(1)$ random variables. For $k \in \mathbb{Z}_{>0}$, define
$$S_k = \sum_{i=1}^k X_i$$
and note that $\mathbb{E}[S_k] = k$.
I was wondering if there is any w …
- 485
2
votes
1
answer
222
views
Maximum of sums of iid $X_i$'s where $X_i$ is the difference of two exponential r.v
Given $X_i = A_i - B_i$ where $A_i\sim \text{ Exp}(\alpha)$ and $B_i \sim \text{ Exp}(\lambda)$. Define $S_k = \sum_{i=1}^k X_i$ with $S_0 = 0$, and
$$M_n = \max_{1\leq k \leq n} S_k.$$
Is it possibl …
- 485