  A step-reinforced random walk is a discrete-time non-Markovian process with
long range memory. At each step, with a fixed probability p, the positively
step-reinforced random walk repeats one of its preceding steps chosen uniformly
at random, and with complementary probability 1-p, it has an independent
increment. The negatively step-reinforced random walk follows the same
reinforcement algorithm but when a step is repeated its sign is also changed.
Strong laws of large numbers and strong invariance principles are established
for positively and negatively step-reinforced random walks in this work. Our
approach relies on two general theorems on invariance principle for martingale
difference sequences and a truncation argument. As by-products of our main
results, the law of iterated logarithm and the functional central limit theorem
are also obtained for step-reinforced random walks.
