# Sum of independent random walks

Given two independent random walks $$S$$ and $$S'$$ with different distributions for the random variables $$X_1$$ and $$X_1'$$, I am interested in studying the conditions that make their sum either a recurrent or transient random walk. Could anyone suggest relevant references?

• The sum of $S$ and $S'$ is again a random walk; why not use standard results for random walks then? Commented Nov 8, 2018 at 9:08

A good criterion is Chung Fuchs Theorem ("On the distribution of values of sums of independent random variables" 1951, it is explained in "Probability Theory, example and application" by Durrett). It tells you that S is transient iff $$Re (1-E[e^{itX_1}])^{-1}$$ is integrable near 0.

As explained by M. Dus, in the case of $$L^1$$ walks looking at expectation gives the answer. This can be recovered using Chung Fuchs Theorem.

If we remove the hypothesis of finite expectation then the situation is more exotic :

• The sum of two transient walk can be recurrent ($$X_1=1, X'_1=-1$$)
• The sum of a transient walk and a recurrent walk can be recurrent ($$X_1=1, X'_1\sim Cauchy$$)
• If the laws are symmetric then the sum of a transient walk and a recurrent walk is always transient (it is a consequence of Chung Fuchs Theorem)
• The most interesting case is the last one : Is it possible for the sum of two independent recurrent walker to be transient ? (another formulation : two independent recurrent walker who meet only a finite number of time). The answer is yes but the construction I know is not that simple. I will sketch it below.

Construction of two independent recurrent walks whose sum is transient : Take Y your favorite symmetric law on $$\mathbb Z$$ such that the walk with steps with the same law than $$Y$$ is transient (for example $$P(Y=n) = \alpha (1+|n|)^{-\frac{3}{2}}$$). Now let $$A_k$$ be an increasing sequence to be adequalty chosen latter and define $$P(X_1=n)=c\sum_{k\geqslant 1} 1_{A_{2k}\leqslant|n| $$P(X'_1=n)=c'\sum_{k\geqslant 1} 1_{A_{2k-1}\leqslant|n| Then

• $$1-E[e^{it(X_1+X'_1)}]$$ has the same behavior in $$0$$ than $$1-E[e^{itX_1}]+1-E[e^{itX'_1}]$$ which has the same behavior than $$1-E[e^{itY}]$$ so that the sum of the two walks is transient like the walks of steps with the same law than $$Y$$.
• Now we choose the $$A_k$$ so that the two walks are recurrent. For all $$k$$, consider $$S^k$$ the walk with steps with the same law than $$X_1 1_{|X_1|. Then $$S^k$$ is recurrent (bounded steps with 0 expectation) thus we can choose $$A_{2k}$$ so large such that with probability $$ the first time that there is a step of size $$> A_{2k-1}$$ in $$S$$ (this is a geometric of parameter less than $$cP(|Y|>=A_{2k})=A_{2k})/P(|Y|=A_2)$$) is after the first time $$S^k$$ returns to $$0$$. But on this event $$S$$ returns to $$0$$ before making its first jump bigger than $$A_{2k-1}$$. You can construct recursively in this way the $$A_k$$'s (look at $$S$$ for even $$k$$ and $$S'$$ for odd $$k$$). Then by Borel-Cantelli there will be for infinitely many $$k$$ where $$S$$ and $$S'$$ returns to $$0$$ and thus they are recurrent.

This is too long for a comment.

You did not say what moment finitess you are assuming. I assume that $$S$$ and $$S'$$ have at least exponential moments below.

As explained by Mateusz Kwaśnick, the sum of $$S$$ and $$S'$$ is just the random walk driven by $$X_1+X_1'$$. So you can apply results for random walks to $$X_1+X_1'$$.

Now it depends on what group your random walk is defined. As you talk about sums, I understand that you are working in an abelian group. Is it $$\mathbb{Z}$$, $$\mathbb{Z}^2$$, or more generally $$\mathbb{Z}^d$$ ?

If you are working in $$\mathbb{Z}$$ or $$\mathbb{Z}^2$$, the random walk is transient if and only if it is non-centered. So $$S+S'$$ is transient if and only if $$\mathbb{E}(X_1+X_1')\neq 0$$ if and only if $$\mathbb{E}(X_1)\neq -\mathbb{E}(X_1')$$. If you are working in $$\mathbb{Z}^d$$, $$d\geq 3$$, then the random walk is always transient.