The narrowest tube of a recurrent random walk

Abstract

Let X1, X2, ... be i.i.d. random variables with P(X1=+1)=P(X1 =−1)=1/2. Put S0=0, Sn=X1+...+Xn (n≧1). Our aim is to investigate the a.s. behavior of $$U(a_N ,N) = \mathop {\min }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } j\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } N - a_N } \mathop {\max }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } a_N } \left| {S_{j + i} - S_j } \right|$$ and $$V(a_N ,N) = \mathop {\min }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } j\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } N - a_N } \mathop {\max }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } a_N } \left| {S_{j + i} } \right|$$ . It is shown that for aN=[c log N] both U(aN, N) and V(aN, N) are a.s. constant for large N, except for certain values of c, when U and V can take two values for large N. The result is extended for recurrent random walk too.