Abstract:
New results will be presented about asymptotics of large deviations probabilities
for random walk in random environment with cookies (RWREwC). Let us denote the process. Let
$$ \vec{p} = \{p(i)\}_{i \in \mathbb{Z}}, \quad \vec{p_1} = \{p_1(i)\}_{i \in \mathbb{Z}} $$
be independent sequences of independent identically distributed random variables taking values in
$(0, 1)$. Let a sequence $\{X_n\}_{n \in \mathbb{N}}$ be denoted as follows:
$$
\mathbf{P}(X_{n+1} = 1|\vec{p}, \vec{p_1}, X_1, . . . , X_n, S_n = i, D_n(i) = 0) = p_1(i), \\
\mathbf{P}(X_{n+1} = -1|\vec{p}, \vec{p_1}, X_1, . . . , X_n, S_n = i, D_n(i) = 0) = 1 - p_1(i), \\
\mathbf{P}(X_{n+1} = 1|\vec{p}, \vec{p_1}, X_1, . . . , X_n, S_n = i, D_n(i) > 0) = p_1(i), \\
\mathbf{P}(X_{n+1} = -1|\vec{p}, \vec{p_1}, X_1, . . . , X_n, S_n = i, D_n(i) > 0) = 1 - p_1(i).
$$
Here
$$ S_0 := 0,\: S_n := \sum_{i=1}^{n}X_i,\quad D_n(i) := \sum_{j=1}^{n}I(S_{j-1} = i, X_j = -1),\quad n \in \mathbb{N},\: i \in \mathbb{Z}. $$
The process $\{S_n\}_{n \ge 0}$ is called random walk in random environment with cookies. It generalizes the
classical model of random walk in random environment (RWRE) inroduced by F. Solomon in [1]. If
the laws of distribution of $p_1(0)$ and $p(0)$ are the same, then the models of RWRE and RWREwC
are coincide. Limit theorems for RWRE were obtained in [2]. Precise asymptotics of large deviations
probabilities for the first moment RWRE reaching level $n$ were obtained in [3] by the author.
By definition, put
$$
T_0 := 0,\quad T_n := \min\{k \in \mathbb{N} : S_k = n\},\quad n \in \mathbb{N}.
$$
The results on precise asymptotics of $\mathbf{P}(T_n = k)$ in the RWREwC model will be presented.
References
Solomon F., “Random walks in a random environment”, The Annals of Probability, 3:1 (1975), 1–31
Kesten H., Kozlov M., Spitzer F., “A limit law for random walk in a random environment”, Compositio mathematica, 30:2 (1975), 145–168
Bakai G. A., “O bolshikh ukloneniyakh momenta dostizheniya dalekogo urovnya sluchainym bluzhdaniem v sluchainoi srede”, Diskretnaya matematika, 34:4 (2022), 3–13
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