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2 votes
0 answers
114 views

Asymptotic Independence of random walks from increments?

Suppose we have two random walks $(S_n:n\geq 1)$ and $(T_n:n\geq 1)$ building from independent identically distributed increment vectors $\{(X_k,Y_k):k\geq 1\}$, i.e. $S_n=\sum_{k=1}^n X_k, T_n=\sum_{...
4 votes
1 answer
187 views

Asymptotics of a quotient related to a simple random walk

Let $\lambda_0 < \lambda_1$ and $\lambda_0 \lambda_1 > 1$ (i.e. at least $\lambda_1 > 1$). Further, let $S_n$ denote a simple random walk with increment distribution $$ P(X = 0)= P(X= 1) = 1/...
1 vote
0 answers
76 views

Spitzer's condition, a slowly varying function and its behavior

Let $S$ denote a random walk that satisfies Spitzer's condition $$ \frac{1}{n} \sum _{k=1}^n P (S_k > 0 ) \to \rho$$ for some $\rho \in (0,1)$. From the book Regular Variation (Bingham, Goldie, ...
1 vote
0 answers
96 views

Minima of a random walk and an equality for a fraction

Let $S_n := X_1 + \dots + X_n$ denote a random walk with zero mean and finite variance and write $L_n := \min \{ 0, S_1, \dots, S_n\}$. The tail distribution of $L_n$ are well-known and in particular, ...