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We prove a difference analogue of the celebrated Tumura–Hayman–Clunie theorem. Let f be a transcendental entire function, let c be a nonzero constant and let n be a positive integer. If f and 
$\Delta _c^n f$ omit zero in the whole complex plane, then either 
$f(z)=\exp (h_1(z)+C_1 z)$, where 
$h_1$ is an entire function of period c and 
$\exp (C_1 c)\neq 1$, or 
$f(z)=\exp (h_2(z)+C_2 z)$, where 
$h_2$ is an entire function of period 
$2c$ and 
$C_2$ satisfies 
$$ \begin{align*} \bigg(\frac{1+\exp(C_2c)}{1-\exp(C_2 c)}\bigg)^{2n}=1. \end{align*} $$
For any positive integers 
$k_1,k_2$ and any set 
$A\subseteq \mathbb {N}$, let 
$R_{k_1,k_2}(A,n)$ be the number of solutions of the equation 
$n=k_1a_1+k_2a_2$ with 
$a_1,a_2\in A$. Let g be a fixed integer. We prove that if 
$k_1$ and 
$k_2$ are two integers with 
$2\le k_1<k_2$ and 
$(k_1,k_2)=1$, then there does not exist any set 
$A\subseteq \mathbb {N}$ such that 
$R_{k_1,k_2}(A,n)-R_{k_1,k_2}(\mathbb {N}\setminus A,n)=g$ for all sufficiently large integers n, and if 
$1=k_1<k_2$, then there exists a set A such that 
$R_{k_1,k_2}(A,n)-R_{k_1,k_2}(\mathbb {N}\setminus A,n)=1$ for all positive integers n.
Let 
$f(z)=\sum\limits_{j=0}^{\infty} a_j z^j$ be a transcendental entire function and let 
$f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where 
$\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space 
$(\Omega, \mathcal{F}, \mu)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher and Steinhaus entire functions. We prove that, for almost all functions in the family and for any constant C > 1, there exist a constant 
$r_0=r_0(\omega)$ and a set 
$E\subset [e, \infty)$ of finite logarithmic measure such that, for 
$r \gt r_0$ and 
$r\notin E$,
\begin{equation*}|\log M(r, f)- N(r,0, f_\omega)|\le (C/A)^{\frac1{B}}\,\log^{\frac1{B}}\,\log M(r,f) +\log\,\log M(r, f), \qquad a.s.\end{equation*}
where 
$A, B$ are constants, 
$M(r, f)$ is the maximum modulus and 
$N(r, 0, f)$ is the integrated zero-counting function of f. As a by-product of our main results, we prove Nevanlinna’s second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by an integrated counting function, rather than by two integrated counting functions as in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions fω and for any ϵ > 0, there is r0 such that, for 
$r \gt r_0$,
\begin{equation*}T(r, f) \le N(r,0, f_\omega)+\left(\tfrac12+\epsilon\right) \log T(r, f).\end{equation*}
Let 
$({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure 
$\nu $ and let f be a real-valued Hölder continuous function on 
${\mathbb X}$ such that 
$\nu (f) = 0$. Consider the Birkhoff sums 
$S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$, 
$n\geqslant 1$. For any 
$t \in {\mathbb R}$, denote by 
$\tau _t^f$ the first time when the sum 
$t+ S_n f$ leaves the positive half-line for some 
$n\geqslant 1$. By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as 
$ n\to \infty $ of the probabilities 
$ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and 
$ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$. We also establish integral and local-type limit theorems for the sum 
$t+ S_n f(x)$ conditioned on the set 
$\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$
