Krishanu Maulik

For a generalized step reinforced random walk, starting from the origin, the first step is taken according to the first element of an innovation sequence. Then in subsequent epochs, it recalls a past epoch with probability proportional to a regularly varying sequence $\{μ_n\}$ of index $γ>-1$; recalls and repeats the step taken with probability $p$, or with probability $1-p$ takes a fresh step from the innovation sequence. The innovation sequence is assumed to be i.i.d.\ with mean zero. We study the corresponding step reinforced random walk process with linearly scaled time as an r.c.l.l.\ function on $[0, \infty)$. We prove law of large numbers for the linearly scaled process almost surely and in $L^1$ for all possible values of $p$ and $γ$. Assuming finite second moments for the innovation sequence, we obtain interesting phase transitions based on the boundedness of a sequence associated with $\{μ_n\}$. The random walk suitably scaled converges almost surely to a process, which may not be Gaussian, when the sequence is bounded and the convergence is in distribution to a Gaussian process otherwise. This phase transition introduces the point of criticality at $p_c=\frac{γ+1/2}{γ+1}$ for $γ>-\frac12$. For the subcritical regime, the process is diffusive, while it is superdiffusive otherwise. However, for the critical regime, the scaled process can converge almost surely or in distribution depending on the choice of sequence $\{μ_n\}$. Almost sure convergence in the critical regime is new. In the critical regime, the scaling can include many more novel choices in addition to $\sqrt{n \log n}$. Further, we use linear time scale and time independent scales even for the critical regime. We argue the exponential time scale for the critical regime is not natural. All the convergences in all the regimes are obtained for the process as an r.c.l.l.\ function.

We consider a randomized urn model with objects of finitely many colors. The replacement matrices are random, and are conditionally independent of the color chosen given the past. Further, the conditional expectations of the replacement matrices are close to an almost surely irreducible matrix. We obtain almost sure and $L^1$ convergence of the configuration vector, the proportion vector and the count vector. We show that first moment is sufficient for i.i.d.\ replacement matrices independent of past color choices. This significantly improves the similar results for urn models obtained in Athreya and Ney (1972) requiring $L\log_+ L$ moments. For more general adaptive sequence of replacement matrices, a little more than $L\log_+ L$ condition is required. Similar results based on $L^1$ moment assumption alone has been considered independently and in parallel in Zhang (2018). Finally, using the result, we study a delayed elephant random walk on the nonnegative orthant in $d$ dimension with random memory.

The stochastic approximation algorithm is a useful technique which has been exploited successfully in probability theory and statistics for a long time. The step sizes used in stochastic approximation are generally taken to be deterministic and same is true for the drift. However, the specific application of urn models with random replacement matrices motivates us to consider stochastic approximation in a setup where both the step sizes and the drift are random, but the sequence is uniformly bounded. The problem becomes interesting when the negligibility conditions on the errors hold only in probability. We first prove a result on stochastic approximation in this setup, which is new in the literature. Then, as an application, we study urn models with random replacement matrices. In the urn model, the replacement matrices need neither be independent, nor identically distributed. We assume that the replacement matrices are only independent of the color drawn in the same round conditioned on the entire past. We relax the usual second moment assumption on the replacement matrices in the literature and require only first moment to be finite. We require the conditional expectation of the replacement matrix given the past to be close to an irreducible matrix, in an appropriate sense. We do not require any of the matrices to be balanced or nonrandom. We prove convergence of the proportion vector, the composition vector and the count vector in $L^{1}$, and hence in probability. It is to be noted that the related differential equation is of Lotka–Volterra type and can be analyzed directly.

We consider a branching random walk on a multitype (withQtypes of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of typeQhave regularly varying tails of index $\alpha$ , while the other types of particles have lighter tails than the particles of typeQ. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in thenth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in thenth generation.

Stochastic approximation algorithm is a useful technique which has been exploited successfully in probability theory and statistics for a long time. The step sizes used in stochastic approximation are generally taken to be deterministic and same is true for the drift. However, the specific application of urn models with random replacement matrices motivates us to consider stochastic approximation in a setup where both the step sizes and the drift are random, but the sequence is uniformly bounded. The problem becomes interesting when the negligibility conditions on the errors hold only in probability. We first prove a result on stochastic approximation in this setup, which is new in the literature. Then, as an application, we study urn models with random replacement matrices. In the urn model, the replacement matrices need neither be independent, nor identically distributed. We assume that the replacement matrices are only independent of the color drawn in the same round conditioned on the entire past. We relax the usual second moment assumption on the replacement matrices in the literature and require only first moment to be finite. We require the conditional expectation of the replacement matrix given the past to be close to an irreducible matrix, in an appropriate sense. We do not require any of the matrices to be balanced or nonrandom. We prove convergence of the proportion vector, the composition vector and the count vector in $L^1$, and hence in probability. It is to be noted that the related differential equation is of Lotka-Volterra type and can be analyzed directly.

We consider a branching random walk on a multi($Q$)-type, supercritical Galton-Watson tree which satisfies Kesten-Stigum condition. We assume that the displacements associated with the particles of type $Q$ have regularly varying tails of index $\alpha$, while the other types of particles have lighter tails than that of particles of type $Q$. In this article, we derive the weak limit of the sequence of point processes associated with the positions of the particles in the $n^{th}$ generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process (SScDPPP) using the tools developed in \cite{bhattacharya:hazra:roy:2016}. As a consequence, we shall obtain the asymptotic distribution of the position of the rightmost particle in the $n^{th}$ generation.

Our work aims to study the tail behaviour of weighted sums of the form $\sum_{i=1}^{\infty} X_{i} \prod_{j=1}^{i}Y_{j}$, where $(X_{i}, Y_{i})$ are independent and identically distributed, with common joint distribution bivariate Sarmanov. Such quantities naturally arise in financial risk models. Each $X_{i}$ has a regularly varying tail. With sufficient conditions similar to those used by Denisov and Zwart (2007) imposed on these two sequences, and with certain suitably summable bounds similar to those proposed by Hazra and Maulik (2012), we explore the tail distribution of the random variable $\sup_{n \geq 1}\sum_{i=1}^{n} X_i \prod_{j=1}^{i}Y_{j}$. The sufficient conditions used will relax the moment conditions on the $\{Y_{i}\}$ sequence.

Consider a sequence $\{(X_{i}, Y_{i})\}$ of independent and identically distributed random vectors, with joint distribution bivariate Sarmanov. This is a natural set-up for discrete time financial risk models with insurance risks. Of particular interest are the infinite time ruin probabilities $P\left[\sup_{n \geq 1}\sum_{i=1}^{n} X_i \prod_{j=1}^{i}Y_{j} > x\right]$. When the $Y_{i}$'s are assumed to have lighter tails than the $X_{i}$'s, we investigate sufficient conditions that ensure each $X_{i}$ has a regularly varying tail, given that the ruin probability is regularly varying. This is an inverse problem to the more traditional analysis of the ruin probabilities based on the tails of the $X_{i}$'s. We impose moment-conditions as well as non-vanishing Mellin transform assumptions on the $Y_{i}$'s in order to achieve the desired results. But our analysis departs from the more conventional assumption of independence between the sequences $\{X_{i}\}$ and $\{Y_{i}\}$, instead assuming each $(X_{i}, Y_{i})$ to be jointly distributed as bivariate Sarmanov, a fairly broad class of bivariate distributions.