Lower deviations for branching processes with immigration

Abstract Let left brace upper Y Subscript n Baseline { Y n $\{Y_{n}$ , n greater than or equals 1 right brace n ≥ 1 } $n \geq 1\}$ be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper we study lower deviation probabilities for upper Y Subscript n Y n $Y_{n}$ . More precisely, assuming that k comma n right arrow normal infinity k , n → ∞ $k,n \to \infty$ so that k equals normal o left parenthesis n right parenthesis k = o ( n ) $k={\mathrm{o}} (n)$ , we investigate the asymptotics of double struck upper P left parenthesis upper Y Subscript n Baseline less than or equals k right parenthesis P ( Y n ≤ k ) $\mathbb P(Y_{n} \leq k )$ and double struck upper P left parenthesis upper Y Subscript n Baseline equals k right parenthesis P ( Y n = k