We consider the coupon collection problem, where each coupon is one of the types 1,…,s with probabilities given by a vector 𝒑. For specified numbers r 1,…,r s , we are interested in finding 𝒑 that minimizes the expected time to obtain at least r i type-i coupons for all i=1,…,s. For example, for s=2, r 1=1, and r 2=r, we show that p 1=(logr−log(logr))∕r is close to optimal.

The explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius r ess(P |𝓁²(𝜋)) of P |𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−N N lim supi→+∞(P(i,i+m)P *(i+m,i)1∕2<1. Moreover, r ess(P |𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

An extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer N let (X N (r))r be a Markov chain with the same finite state space S and transition matrix ΠN =I+d N B N , where I is the unit matrix, Q a generator matrix, (B N )N a sequence of matrices, limN℩∞ c N = limN→∞d N =0 and limN→∞ c N ∕d N =0. Suppose that the limits P≔limm→∞(I+d N Q)m and G≔limN→∞ P B N P exist. If the sequence of initial distributions P X N (0) converges weakly to some probability measure μ, then the finite-dimensional distributions of (X N ([t∕c N ))t≥0 converge to those of the Markov process (X t )t≥0 with initial distribution μ, transition matrix PetG and limN→∞(I+d N Q+c N B N )[t∕c N]