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Let $\tau_{x,y}$ be the first hitting time of $y$ starting from $x$ at time $0$. Let $\tau_y:=\tau_{0,y}$. Then the conditions $h^+_N < h^-_N$ and $c^-_N < e^+_N$ can be rewritten as $\tau_N<\tau_{-N} …

One way to do this is as follows. We have to show that $$P(M_n\ge x|S_n=0)\to1$$ (as $n\to\infty$) if $x=o(\sqrt n)$, where $S_n$ is the position of the walk at time $n$ and $M_n:=\max_{0\le k\le n}S …

The expectation of $S$ is indeed $0$. This follows by the optional stopping theorem; see e.g. THM 29.11 with $X=W$, $S=0$, and $T:=\inf\{t\ge0\colon W_t\notin(-1,0.1)\}$. Alternatively and more specif …

Let $$p(a):=P\big(\max_{n\ge0} S_n\le a\big).$$ Assume that $c_3:=E|X_1-EX_1|^3<\infty$. By the improvement by Sakhanenko of Lemma 8 by S. Nagaev (the improvement consisting in removing an extra facto …

Recall that a random walk (or a Markov chain in general) is called recurrent if it almost surely (a.s.) returns to the initial state infinitely often. We will show that in our case the walk is recurre …

Recall that a random walk (or a Markov chain in general) is called recurrent if it almost surely (a.s.) returns to the initial state infinitely often. The previous answer, which showed that the walk ( …

$\newcommand\ep\epsilon$The OP, who wanted more elementary arguments, agreed that the cases $t<1/2$ and $t>1$ would be enough. Therefore I am providing the third answer to the question, with more elem …

This can be done by the reflection principle. Also, one can use Theorem 0.6, which implies $$P(\tau_0^+>k)=\tfrac1k\,E|S_k|.$$ By the central limit theorem and uniform integrability, for $k\to\infty$, …

Let $m:=\lfloor\log_2 k\rfloor$, so that $2^m\le k<2^{m+1}$. Then the probability in question is $$ \begin{aligned} P_k&:=P(|W_n|\le\sqrt n\ \forall n\le k) \\ &\le P(|W_{2^j}|\le2^{j/2}\ \forall j\i …

By obvious rescaling, without loss of generality $\sigma=1$. Assume also that $c_3:=E|X_1|^3<\infty$. By the well-known result about the rate of convergence in boundary value problems for random walks …

Your boundary conditions do not correspond to reflective boundaries. Your $P(n,t)$ is the probability that, starting at time $0$ at some point $x_0$ in the set $I:=\{2,\dots,N-1\}$, the random walker …

$\newcommand\R{\mathbb R}\newcommand\Z{\mathbb Z}\newcommand\De\Delta$This follows almost immediately from the multidimensional local limit theorem (MLLT) proved by Meĭzler, D. G.; Parasyuk, O. S.; Rv …

By the de Moivre–Laplace theorem, \begin{equation} P(S_n=k)=P(B_n=(n+k)/2)\sim\frac1{\sqrt{\pi n/2}}\,\exp\{-k^2/(2n)\}, \end{equation} where $B_n$ is a random variable with the binomial distributio …

Let $X_1,X_2,\dots$ be iid random vectors each uniformly distributed on $S^{d-1}$. Let $S_n:=\sum_1^n X_i$. By the symmetry, $EX_1=0$. Also, $1=|X_1|^2=\sum_{j=1}^d X_{1j}^2$, where $X_1=(X_{11},\dots …

Such results were obtained by A. A. Borovkov and his students. See e.g. Borovkov, A. A. Estimates in the invariance principle. (Russian) Dokl. Akad. Nauk SSSR 206 (1972), 1037–1039. Borovkov, A. A. …