Abstract
Let $$X_1, X_2, \ldots $$
X
1
,
X
2
,
…
be i.i.d. copies of some real random variable X. For any deterministic $$\varepsilon _2, \varepsilon _3, \ldots $$
ε
2
,
ε
3
,
…
in $$\{0,1\}$$
{
0
,
1
}
, a basic algorithm introduced by H.A. Simon yields a reinforced sequence $$\hat{X}_1, \hat{X}_2 , \ldots $$
X
^
1
,
X
^
2
,
…
as follows. If $$\varepsilon _n=0$$
ε
n
=
0
, then $$ \hat{X}_n$$
X
^
n
is a uniform random sample from $$\hat{X}_1, \ldots , \hat{X}_{n-1}$$
X
^
1
,
…
,
X
^
n
-
1
; otherwise $$ \hat{X}_n$$
X
^
n
is a new independent copy of X. The purpose of this work is to compare the scaling exponent of the usual random walk $$S(n)=X_1+\cdots + X_n$$
S
(
n
)
=
X
1
+
⋯
+
X
n
with that of its step reinforced version $$\hat{S}(n)=\hat{X}_1+\cdots + \hat{X}_n$$
S
^
(
n
)
=
X
^
1
+
⋯
+
X
^
n
. Depending on the tail of X and on asymptotic behavior of the sequence $$(\varepsilon _n)$$
(
ε
n
)
, we show that step reinforcement may speed up the walk, or at the contrary slow it down, or also does not affect the scaling exponent at all. Our motivation partly stems from the study of random walks with memory, notably the so-called elephant random walk and its variations.
Further scaling exponents of random walks in random sceneries