Randomly Weighted Self-normalized L\'evy Processes

Let $(U_t,V_t)$ be a bivariate L\'evy process, where $V_t$ is a subordinator and $U_t$ is a L\'evy process formed by randomly weighting each jump of $V_t$ by an independent random variable $X_t$ having cdf $F$. We investigate the asymptotic distribution of the self-normalized L\'evy process $U_t/V_t$ at 0 and at $\infty$. We show that all subsequential limits of this ratio at 0 ($\infty$) are continuous for any nondegenerate $F$ with finite expectation if and only if $V_t$ belongs to the centered Feller class at 0 ($\infty$). We also characterize when $U_t/V_t$ has a non-degenerate limit distribution at 0 and $\infty$.Comment: 32 page

On the Breiman conjecture

Let $Y_{1},Y_{2},\ldots$ be positive, nondegenerate, i.i.d. $G$ random variables, and independently let $X_{1},X_{2},\ldots$ be i.i.d. $F$ random variables. In this note we show that whenever $\sum X_{i}Y_{i}/\sum Y_{i}$ converges in distribution to nondegenerate limit for some $F\in \mathcal{F}$, in a specified class of distributions $\mathcal{F}$, then $G$ necessarily belongs to the domain of attraction of a stable law with index less than 1. The class $\mathcal{F}$ contains those nondegenerate $X$ with a finite second moment and those $X$ in the domain of attraction of a stable law with index $1<\alpha <2$

Asymptotics of nearly critical Galton-Watson processes with immigration

We investigate the inhomogeneous Galton--Watson processes with immigration, where $\rho_n$ the offspring means in the $n^\textrm{th}$ generation tends to 1. We show that if the second derivatives of the offspring generating functions go to 0 rapidly enough, then the asymptotics are the same as in the INAR(1) case, treated by Gy\"orfi et al. We also determine the limit if this assumption does not hold showing the optimality of the conditions.Comment: 25 page

Darling-Kac theorem for renewal shifts in the absence of regular variation

We study null recurrent renewal Markov chains with renewal distribution in the domain of geometric partial attraction of a semistable law. Using the classical procedure of inversion, we derive a limit theorem similar to the Darling-Kac law along subsequences and obtain some interesting properties of the limit distribution. Also in this context, we obtain a Karamata type theorem along subsequences for positive operators. In both results, we identify the allowed class of subsequences. We provide several examples of nontrivial infinite measure preserving systems to which these results apply.Comment: 39 pages, 4 figure