The authors investigate the behaviour of the counting function $N^{\alpha,c,a}_t[0,L] = \sum_{j=-\infty}^{\infty}\Bbb{I}[X^{\alpha,c}_j(t) + u_j\in[0,L]]$ for $t\rightarrow\infty$ and $L\rightarrow\infty$, where $X^{\alpha,c}_j(t)$ are independent symmetric $\alpha$-stable Lévy processes which start from the initial positions $u_j=a(j-\epsilon)$ where $a>0$, $\epsilon \sim {\rm Uniform}[0,1]$ independent on $X^{\alpha,c}_j$ and $j\in\Bbb Z$. The most important tool for achieving results in this paper is an explicit formula for the number variance $V_t^{\alpha,c,a}[L]=\text{Var}(N^{\alpha,c,a}_t[0,L])$. As $V_t^{\alpha,c,a}[L]$ diverges for $\alpha\in(0,1]$ as $L\rightarrow\infty$ one can approximate the statistic $N^{\alpha,c,a}_t[0,L]$ by the Poisson random variable with mean $L/a$. $V_t^{\alpha,c,a}[L]$ is convergent for $\alpha\in(1,2]$ as $L\rightarrow\infty$. The ``Costin-Lebowitz-Soshnikov'' method is used to prove the convergence of $(N_t^{\alpha,c,a}[0,L] - L/a)/\sqrt{V_t^{\alpha,c,a}[L]}$ for $\alpha\in(0,1]$ to a standard normal distribution as $L\rightarrow\infty$. The weak convergence of measures induced on the space $C[0,\infty)$ by continuous approximations of processes $$Z_t^{\alpha,c,a}(s) \coloneq (N_t^{\alpha,c,a}[0,st^{1/\alpha}] - st^{1/\alpha}/a)t^{-1/2\alpha},\;s\geq 0,\; \alpha\in(0,2]$$ for $t\rightarrow\infty$ is proved by the tightness of the measures and by the convergence of finite-dimensional distributions of the $Z_t^{\alpha,c,a}(s)$. The limiting process $G^{\alpha,c,a}(s)$ is centered, Gaussian, with variance $V_1^{\alpha,c,a}[s]$ and with negatively correlated stationary increments. Its covariance structure is similar to that of the fractional Brownian motion. A properly rescaled $G^{\alpha,c,a}(s)$ may be approximated by the fractional Brownian motion with Hurst parameter $(1-\alpha)/2$ for $\alpha\in(0,1)$. \par \{For additional information pertaining to this item see [B. M. Hambly\ and L. Jones, Electron. J. Probab. {\bf 14} (2009), No. 37, 1074--1079; 2506125 ].\} \par REVISED (April, 2010) \prevrevtext The authors investigate the behaviour of the counting function $N^{\alpha,c,a}_t[0,L] = \sum_{j=-\infty}^{\infty}\Bbb{I}[X^{\alpha,c}_j(t) + u_j\in[0,L]]$ for $t\rightarrow\infty$ and $L\rightarrow\infty$, where $X^{\alpha,c}_j(t)$ are independent symmetric $\alpha$-stable Lévy processes which start from the initial positions $u_j=a(j-\epsilon)$ where $a>0$, $\epsilon \sim {\rm Uniform}[0,1]$ independent on $X^{\alpha,c}_j$ and $j\in\Bbb Z$. The most important tool for achieving results in this paper is an explicit formula for the number variance $V_t^{\alpha,c,a}[L]=\text{Var}(N^{\alpha,c,a}_t[0,L])$. As $V_t^{\alpha,c,a}[L]$ diverges for $\alpha\in(0,1]$ as $L\rightarrow\infty$ one can approximate the statistic $N^{\alpha,c,a}_t[0,L]$ by the Poisson random variable with mean $L/a$. $V_t^{\alpha,c,a}[L]$ is convergent for $\alpha\in(1,2]$ as $L\rightarrow\infty$. The ``Costin-Lebowitz-Soshnikov'' method is used to prove the convergence of $(N_t^{\alpha,c,a}[0,L] - L/a)/\sqrt{V_t^{\alpha,c,a}[L]}$ for $\alpha\in(0,1]$ to a standard normal distribution as $L\rightarrow\infty$. The weak convergence of measures induced on the space $C[0,\infty)$ by continuous approximations of processes $$Z_t^{\alpha,c,a}(s) \coloneq (N_t^{\alpha,c,a}[0,st^{1/\alpha}] - st^{1/\alpha}/a)t^{-1/2\alpha},\;s\geq 0,\; \alpha\in(0,2]$$ for $t\rightarrow\infty$ is proved by the tightness of the measures and by the convergence of finite-dimensional distributions of the $Z_t^{\alpha,c,a}(s)$. The limiting process $G^{\alpha,c,a}(s)$ is centered, Gaussian, with variance $V_1^{\alpha,c,a}[s]$ and with negatively correlated stationary increments. Its covariance structure is similar to that of the fractional Brownian motion. A properly rescaled $G^{\alpha,c,a}(s)$ may be approximated by the fractional Brownian motion with Hurst parameter $(1-\alpha)/2$ for $\alpha\in(0,1)$. \prevrevr Michal\ Vyoral \endprevrevtext