## Journal of Applied Probability

- J. Appl. Probab.
- Volume 50, Number 2 (2013), 499-515.

### On the number of turns in reduced random lattice paths

#### Abstract

We consider the tree-reduced path of a symmetric random walk on
ℤ^{d}. It is interesting to ask about the number of turns
*T*_{n}
in the reduced path after *n* steps. This question arises from
inverting the signatures of lattice paths: *T*_{n} gives an upper bound
of the number of terms in the signature needed to reconstruct a
`random' lattice path with *n* steps. We show that, when *n* is large,
the mean and variance of *T*_{n} in the asymptotic expansion have the
same order as *n*, while the lower-order terms are *O*(1). We also
obtain limit theorems for *T*_{n}, including the large deviations
principle, central limit theorem, and invariance principle. Similar
techniques apply to other finite patterns in a lattice path.

#### Article information

**Source**

J. Appl. Probab., Volume 50, Number 2 (2013), 499-515.

**Dates**

First available in Project Euclid: 19 June 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1371648957

**Digital Object Identifier**

doi:10.1239/jap/1371648957

**Mathematical Reviews number (MathSciNet)**

MR3102496

**Zentralblatt MATH identifier**

1316.60065

**Subjects**

Primary: 60F10: Large deviations 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks

**Keywords**

Signature of a path reduced word number of turns

#### Citation

Jiang, Yunjiang; Xu, Weijun. On the number of turns in reduced random lattice paths. J. Appl. Probab. 50 (2013), no. 2, 499--515. doi:10.1239/jap/1371648957. https://projecteuclid.org/euclid.jap/1371648957