public class ProcessSchedulerServlet implements javax.servlet.Servlet {
    Timer timer=new Timer();
    @Override
    public void init(ServletConfig arg0) throws ServletException {
        timer.scheduleAtFixedRate(new TimerTask() {
        public void run() {
            LogProcessorService logProcessorService=new LogProcessorServiceImpl();
            logProcessorService.processPageRequestsLogs();
        }
     }, 60*1000, 120*1000);
}
This is ugly and it doesn't work, anyway. The LogProcessorServiceImpl has properties with @Autowired annotation. These properties are not autowired when this code runs. This may be expected.
The real question is: how to make this run() method work. It seems to me that Spring wants the logProcessorService to be autowired to have properties within LogProcessorServiceImpl autowired, as well.
=== SCENARIO 1 ==============================================================
public void run() {
    final LogProcessorService logProcessorService=null;
    WebApplicationContextUtils.getWebApplicationContext(arg0.getServletContext()).getAutowireCapableBeanFactory().autowireBean(logProcessorService);
    logProcessorService.processPageRequestsLogs();
}
Result: compile time error: Cannot refer to a non-final variable arg0 inside an inner class defined in a different method
=== SCENARIO 2 ==============================================================
@Autowired
LogProcessorService logProcessorService;
public void run() {
    logProcessorService.processPageRequestsLogs();
}
Result: run time error: logProcessorService is null;
==== SOLUTION (from Boris) ======================================================
public class ProcessSchedulerServlet implements javax.servlet.Servlet {
    Timer timer=new Timer();
    @Autowired
    LogProcessorService logProcessorService;
    @Override
    public void init(ServletConfig arg0) throws ServletException {
        final AutowireCapableBeanFactory autowireCapableBeanFactory=WebApplicationContextUtils.getWebApplicationContext(arg0.getServletContext()).getAutowireCapableBeanFactory();
        autowireCapableBeanFactory.autowireBean(this);
        timer.scheduleAtFixedRate(new TimerTask() {
            public void run() {
                logProcessorService.processPageRequestsLogs();
            }
        }, 60*1000, 120*1000);
}